# Zwicky Transient Facility and Globular Clusters: the $gr$-Band Period-
Luminosity Relations for Mira Variables at Maximum Light and Their
Applications to Local Galaxies

Chow-Choong Ngeow Graduate Institute of Astronomy, National Central
University, 300 Jhongda Road, 32001 Jhongli, Taiwan  Jia-Yu Ou Graduate
Institute of Astronomy, National Central University, 300 Jhongda Road, 32001
Jhongli, Taiwan  Anupam Bhardwaj INAF-Osservatorio astronomico di Capodimonte,
Via Moiariello 16, 80131 Napoli, Italy  Josiah Purdum Caltech Optical
Observatories, California Institute of Technology, Pasadena, CA 91125, USA
Ben Rusholme IPAC, California Institute of Technology, 1200 E. California
Blvd, Pasadena, CA 91125, USA  Avery Wold IPAC, California Institute of
Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA

###### Abstract

Based on 14 Miras located in 7 globular clusters, we derived the first
$gr$-band period-luminosity (PL) at maximum light for the large-amplitude Mira
variables using the multi-year light-curve data collected from the Zwicky
Transient Facility (ZTF). Since Miras are red variables, we applied a color-
term correction to subsets of ZTF light curves, and found that such
corrections do not have a large impact on period determinations. We applied
our derived PL relations to the known extragalactic Miras in five local
galaxies (Sextans, Leo I, Leo II, NGC6822 and IC1613), and determined their
Mira-based distances. We demonstrated that our PL relations can be applied to
short-period ($\lesssim 300$ days) Miras, including those in the two most
distant galaxies (NGC6822 and IC1613) in our sample even when only a portion
of the light-curves around maximum light have detections. We have also shown
that the long-period extragalactic Miras do not follow the PL relations
extrapolated to longer periods. Hence, our derived PL relations are only
applicable to the short-period Miras, which will be discovered in abundance in
local galaxies within the era of Vera C. Rubin Observatory’s Legacy Survey of
Space and Time.

††software: astropy (Astropy Collaboration et al., 2013, 2018, 2022), dustmaps
(Green, 2018), gatspy (VanderPlas & Ivezić, 2015), Matplotlib (Hunter, 2007),
NumPy (Harris et al., 2020), SciPy (Virtanen et al., 2020), statsmodels
(Seabold & Perktold, 2010).

##  1 Introduction

Mira variables (hereafter Miras, see Mattei, 1997, for a general review) are
pulsating asymptotic giant branch (AGB) stars with periods longer than 100
days. Miras obey a period-luminosity (PL) relation, especially in the $K$-band
or the bolometric magnitudes, because Miras are cool supergiants with
radiation peaks in the near-infrared (NIR). Indeed, the majority of the PL
relations derived, or calibrated (that is, calibrating the zero-point of the
PL relation by fixing the slope), in the literature were in the $JHK$-band
(mainly in the $K$-band) and/or being converted to bolometric magnitudes (for
examples, see Glass & Evans, 1981; Glass & Feast, 1982; Feast, 1984; Reid et
al., 1988; Feast et al., 1989; Hughes & Wood, 1990; Groenewegen & Whitelock,
1996; Bedding & Zijlstra, 1998; Whitelock & Feast, 2000; Glass & Lloyd Evans,
2003; Rejkuba, 2004; Soszyński et al., 2005; Feast et al., 2006; Whitelock et
al., 2008; Matsunaga et al., 2009; Tabur et al., 2010; Ita & Matsunaga, 2011;
Yuan et al., 2017, 2018; Bhardwaj et al., 2019; Grady et al., 2019; Ita et
al., 2021; Andriantsaralaza et al., 2022; Sun et al., 2022; Sanders, 2023).
Some of these studies also included the derivation of period-Wesenheit (PW)
relations, or the addition of a color-term for a period-luminosity-color (PLC)
relation. At wavelengths shorter than $J$-band, PL (and PLC) relations have
been derived in various optical bands (including $I$-band and Gaia bands, Ita
& Matsunaga, 2011; Bhardwaj et al., 2019) in addition to the NIR bands, as
well as a single-band PL relation in photographic $m_{pg}$-band (van den
Bergh, 1984) and $I$-band (Ou & Ngeow, 2022). Beyond $K$-band, several works
have also derived the mid-infrared PL relations (Glass et al., 2009; Matsunaga
et al., 2009; Riebel et al., 2010; Ita & Matsunaga, 2011; Iwanek et al.,
2021a). Finally, based on a comprehensive multi-bands analysis for Miras in
the Large Magellanic Cloud (LMC), Iwanek et al. (2021b) presented a set of
synthetic PL relations in 42 bands ranged from $0.37\mu$m to $25.5\mu$m.

In light of the recent “Hubble Tension” (for general reviews, see Verde et
al., 2019; Di Valentino et al., 2021; Freedman, 2021; Shah et al., 2021; Riess
et al., 2022), Miras offer an independent route to determine the $H_{0}$ via
the local distance scale ladder, with several advantages being mentioned in
Whitelock (2013), Macri (2017), Huang et al. (2018, 2020), and Sanders (2023).
On the other hand, Miras are long-period variables implying the long-term
monitoring of extragalactic Miras using space-based telescopes, such as the
James Webb Space Telescope (JWST), might not be trivial (in terms of
scheduling, proposals competitions, etc). On the ground, the 10 years survey
of the Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST,
Ivezić et al., 2019) will naturally provide such a long baseline suitable for
Miras. Hence, in combination with its photometric depths, LSST is expected to
detect numerous Miras in various types of nearby galaxies (Macri, 2017).
Indeed, Miras are included in the Roadmap for the LSST Transients and Variable
Stars group (Hambleton et al., 2022) as one of the scientific targets/goals of
the survey.

Our aim of this work is to derive the optical band PL relations at maximum
light by using the homogeneous light curve data collected from the Zwicky
Transient Facility (ZTF, Bellm et al., 2019a; Graham et al., 2019; Dekany et
al., 2020) for Miras located in the globular clusters. Our work would be
similar to the work of Menzies & Whitelock (1985) and Feast et al. (2002), but
using the most up-to-date and homogeneous globular cluster distances as given
in Baumgardt & Vasiliev (2021). We choose to derive the PL relations at
maximum light because their dispersions were found to be smaller than the mean
light counterparts (Kanbur et al., 1997; Bhardwaj et al., 2019; Ou & Ngeow,
2022; Ou et al., 2023). Though not an optimal choice, the derived optical band
PL relations will still be valuable in the era of LSST, because the six
filters set employed by LSST ($ugrizy$) do not extend to the NIR. In this
manuscript, Section 2 describes the sample of Miras in globular clusters and
their extracted ZTF light curves. We then refined their periods in Section 3,
followed by the derivation of PL relations at maximum light in Section 4. We
applied our derived PL relations to a number of local group galaxies in
Section 5. We concluded our work in Section 6 together with a brief
discussion.

##  2 Sample and ZTF Light Curves

We started the compilation of Miras in globular clusters from the “Updated
Catalog of Variable Stars in Globular Clusters” (Clement et al., 2001;
Clement, 2017, hereafter the Clement’s Catalog), and only selected variable
stars of “M” type. V1 and V4 in Palomar 4 was marked as “SR” (semi-regular) in
Clement’s Catalog, however they were reclassified as Mira in Grady et al.
(2019). Hence, they were added to the compilation. After excluding known or
suspected foreground objects, and V7 in Terzan 5 (because this Mira was found
to be a non-member based on radial velocity measurement, see Origlia et al.,
2019), there were 23 Miras located in 10 globular clusters observable with ZTF
($\delta_{J2000}>-30^{\circ}$).

Figure 1: Distributions of color-coefficients, $C_{m}$, retrieved from the PSF
catalogs for our sample of Miras. Upper and lower panels are for the $g$\- and
$r$-band, respectively.

For this initial sample of globular clusters Miras, we extracted the ZTF
$gri$-band time-series photometric data from the PSF (point-spread function)
catalogs, produced from a dedicated reduction pipeline (Masci et al., 2019),
using a $1\arcsec$ search radius. The PSF catalogs included those available
from the ZTF Data Release (DR) 16 and the ZTF partner survey data111ZTF
observations were divided into three parts, one of them being the partnership
survey (for a further details, see Bellm et al., 2019a). collected until 2023
March 31. We first visually inspected the ZTF light curves and remove those
Miras with small number of data-points, or the data-points only sampling a
small portion of the light curves within a single pulsating cycle. This step
removed V4 in NGC 6553, V12 and V13 in NGC 6638, V5 and V6 in Terzan 5, V1 in
Terzan 12. Three Miras in Terzan 5 either lack $g$-band or only contain very
few $g$-band data points, and were subsequently removed because their single
$r$-band ZTF light curves cannot be used to estimate colors (see next
section). Hence, there are 14 Miras left in 7 globular clusters in the sample.

We have also excluded the $i$-band light curves, because there are 7 Miras in
our sample that do not have any ZTF $i$-band data. For the remaining Miras,
the numbers of data-points for the $i$-band light curves are in general
smaller than other two filters,222After excluding light curves with null data,
the averaged numbers of data points per light curves in the $gri$-band are
242.8, 821.3, and 193.6, respectively. and many of them only cover a portion
of the light curve in a single pulsation cycle, preventing them to be analyzed
further.

##  3 Color-Correction on Light Curves and Periods Refinement

Figure 2: Left Panel: Color curve for one of the Mira in our sample
constructed using equation (1). The “MJD” (modified Julian date) for the
$(g-r)$ data-points are at the mid-point of the $gr$-band MJDs. The magenta
circles and green triangles are for the pairs of $gr$-band data-points taken
in the same night ($\Delta MJD<0.5$ days) and taken on different nights up to
$0.02P_{n}$ days, respectively. Crosses are rejected data-points either with
errors larger than 0.35 or with anomaly $C_{m}$ values. Right Panel: The
$gr$-band light curves for the same Miras as in the left panel. Open and
filled circles in the $gr$-band light curves are those without and with the
color-term corrections, respectively. Note that the color-term corrections did
not apply to all $gr$-band data-points, only for those with the $(g-r)$
counterparts. For clarity, error-bars are omitted in the right panels.

As described in Masci et al. (2019), PSF photometry in ZTF catalogs were
calibrated via $m=m_{ZTF}+ZP_{m}+C_{m}(g-r)$, where $m=\\{g,\ r\\}$ is the
calibrated magnitudes in the Pan-STARRS1 (Chambers et al., 2016; Magnier et
al., 2020) AB magnitude system, and $m_{ZTF}$ represents the ZTF instrumental
magnitudes. For non-varying sources, the $(g-r)$ colors can be obtained from
the Pan-STARRS1 catalog. For variable stars such as Miras, however, the time-
dependent colors have to be known a priori. This is because the $gr$-band
observations in ZTF are not simultaneous or near-simultaneous (Bellm et al.,
2019b). Figure 1 shows the histograms for the color-coefficient, $C_{m}$, with
medians of $-0.053$ and $0.096$ in the $gr$-band, respectively (the
corresponding modes are $-0.052$ and $0.091$). Since Miras are very red
variable stars, the calibrated light curves should include the $C_{m}(g-r)$
color-terms.

Given the long period nature of Miras, the $gr$-band photometry taken within
the same night, or within $\sim 0.02P$ (where $P$ is the pulsation period),
can be treated as “near-simultaneous”. Hence, the time-dependent colors can be
obtained via the following equation (Ngeow, 2022):

$\displaystyle(g-r)$ $\displaystyle=$
$\displaystyle\frac{(g_{ZTF}+ZP_{g})-(r_{ZTF}+ZP_{r})}{1-C_{g}+C_{r}}.$ (1)

We paired up the $gr$-band data-points that are closest in time, up to a
threshold of $\Delta MJD<0.02P_{n}$ (where $MJD$ is the modified Julian date),
to construct the $(g-r)$ colors. The period $P_{n}$ was determined using the
LombScargleMultiband module (VanderPlas & Ivezić, 2015), available from the
astroML/gatspy package,333https://github.com/astroML/gatspy. on all of the
$gr$-band light curves without the color-terms (hence the subscript n means no
color-terms). Left panel of Figure 2 shows an example of the color-curve
constructed using equation (1), while the right panels of Figure 2 present the
$gr$-band light curves without (open circles) and with (filled circles) the
inclusion of the color-terms. In general, the pairs of $gr$-band data-points
separated by more than a night (green triangles in the left panel of Figure 2)
closely resemble those taken within the same night, justifying our assumption
that the colors can be constructed for Miras even if the data-points in two
bands were taken from different nights.

Figure 3: Comparison of the multi-band Lomb-Scargle (LS) periodograms for NGC6642 V1 based on the light curves with (upper panel) and without (lower panel) the corrections of color-terms. |  |
---|---|---
Figure 4: Left Panel: The multi-band LS periodograms based on the ZTF light
curves, at which the highest peaks corresponding to $P_{n}$ (upper panel) and
$P_{c}$ (lower panel). The vertical dashed line indicates the literature
period of $P_{l}=156$ days. Middle Panel: Folded $r$-band light curves with
our determined period $P_{n}$ (upper panel) and literature period $P_{l}$
(lower panel). The filled and open circles are data-points with and without
color-terms correction, respectively. Right Panel: ZTF light curves and color
curve for NGC6638 V15, where the symbols are same as the right panels of
Figure 2.

We have also determined the periods based on the subsets of $gr$-band light
curve data-points that can be used to calculate the color-terms. When
comparing the periods found in these subsets of light curves, the largest
percentage difference is $0.1\%$ between the light curves with and without the
color-terms correction, suggesting the color-terms do not have a great impact
on the determined periods (due to large amplitudes of Miras). The periods
based on the subsets of light curves corrected for the color-terms are denoted
as $P_{c}$. In general, $P_{c}$ is also in good agreement with $P_{n}$, and
the percentage differences between them are less than 1%. The only exception
is NGC6642 V1, for which the full set of $gr$-band light curves, uncorrected
for color-terms, give $P_{n}=131.1$ days. In contrast, the subset of color-
corrected light curves returned a period of $P_{c}=204.6$ days. Figure 3 shows
that the Lomb-Scargle (LS) periodograms for this Mira have similar peaks at
both periods, suggesting aliasing is affecting its light curves. We adopted
$P_{n}=131.1$ days for NGC6642 V1 because it is closer to the literature
period of 127 days. We have also adopted $P_{n}$ as the final periods for
other Miras in our sample, because the total number of data-points for full
set of $gr$-band light curves are $\sim 2$ to $\sim 5$ times more than the
subsets of data-points for determining colors, and have a longer time-span
(hence, reducing the impact of aliasing). Since $P_{n}$ and $P_{c}$ differ by
less than 1%, errors on the adopted $P_{n}$ are most likely at the 1%-level.

Finally, with the exception of NGC6638 V15, the percentage differences between
our determined $P_{n}$ and the literature periods ($P_{l}$) varied from
$0.04\%$ to $6.28\%$. In all cases, light curves folded with $P_{n}$ exhibit a
smaller scatter, justifying our determined periods are robust. For NGC6638
V15, the literature period of $P_{l}=156$ days corresponds to the second
highest peaks of the LS periodograms (see left panels of Figure 4).
Nevertheless, our periods of $P_{n}=274.8$ days can fold the ZTF light curves
better than the literature periods, as demonstrated in the middle panel of
Figure 4. For completeness, the right panels of Figure 4 present the $gr$-band
light curves and the $(g-r)$ color curve for this Mira.

##  4 The PL Relations at Maximum Light

Figure 5: ZTF light curves for two Miras after corrected for the color-terms
(open circles). The filled black squares are the determined magnitudes at
maximum light, using the method described further in the text. Left panels
show a Mira that only two $m^{i}_{X}$ can be determined, while the right
panels are for a Mira with a total of 10 determined $m^{i}_{X}$.

Light curves of Miras are well-known to exhibit cycle-to-cycle variations (for
some exemplar light curves, see Iwanek et al., 2022). Therefore, Bhardwaj et
al. (2019) took the averages of the brightest 10% of the light curves as an
estimate of magnitudes at maximum light, $m_{X}$. While Ou & Ngeow (2022)
divided the light curves to different pulsating cycles, and took an averaged
for $m_{X}$ found in the pulsating cycles. We followed the approach laid out
in Ou & Ngeow (2022) to determine the magnitudes at maximum light for our
sample of globular cluster Miras. Briefly, we divided the ZTF $gr$-band light
curves to a number of pulsation cycles according to the $P_{n}$ determined in
previous section. For $i^{\mathrm{th}}$ pulsation cycle, we fit a sinusoidal
function to determine $m_{X}^{i}$ if there are more than 10 data-points within
the pulsation cycle and contain data-points around the maximum light (see
Figure 5 for two examples). In general, there are 2 to 10 pulsating cycles
which satisfied our selection criteria for determining $m_{X}^{i}$, hence we
took a mean as the final adopted $m_{X}$ for our sample of Miras. The standard
deviations on $m_{X}$, $\sigma_{X}$, were calculated based on small-number
statistics (Dean & Dixon, 1951; Keeping, 1962, p. 202). The values of $m_{X}$
and $\sigma_{X}$, together with periods determined in previous section, are
summarized in Table 1. In general, $\sigma_{X}$ are in the order of $\sim 0.5$
mag. This in turns reflect the fact that light curves for Miras are varying
from cycle-to-cycle, resulting the magnitudes at maximum light fluctuate at
the $\sim 0.5$ mag level.444In general, amplitudes for shorter wavelengths
light curves (such as $gr$-band) would be larger than the $I$-band or
$JHK$-band light curves (Iwanek et al., 2021b, their Figure 2). Hence, it is
expected such fluctuations at maximum light would also be larger for shorter
wavelengths light curves.

Table 1: Observed Properties of the Miras in Globular Clusters Studied in This Work. Mira |  $P_{l}$aaPublished period as given in the literature. (days) |  $P_{n}$ (days) |  $P_{c}$ (days) | $N_{g}$ | $N_{r}$ |  $N_{c}$bb$N_{c}$ is the number of data-points for constructing the color curves (i.e., pairs of $gr$-band photometry separated within $0.02P_{n}$ days); $N_{X}$ is the number of measured magnitudes at maximum light (see text for more details). |  $N_{X}$bb$N_{c}$ is the number of data-points for constructing the color curves (i.e., pairs of $gr$-band photometry separated within $0.02P_{n}$ days); $N_{X}$ is the number of measured magnitudes at maximum light (see text for more details). | $g_{X}$ | $\sigma_{g}$ | $r_{X}$ | $\sigma_{r}$ |  $D$ (kpc)ccDistance of the host globular clusters adopted from Baumgardt & Vasiliev (2021). |  $E$ddReddening value returned from the Bayerstar2019 3D reddening map (Green et al., 2019) at the location of the Miras in the globular clusters.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
NGC6356 V1 | 230.6 | 226.7 | 227.0 | 83 | 423 | 61 | 3 | 15.377 | 0.353 | 13.941 | 0.397 | $15.66\pm 0.92$ | $0.366\pm 0.002$
NGC6356 V3 | 220.0 | 220.4 | 220.4 | 229 | 732 | 100 | 3 | 14.778 | 0.397 | 13.426 | 0.487 | $15.66\pm 0.92$ | $0.366\pm 0.002$
NGC6356 V5 | 219.8 | 220.6 | 220.0 | 256 | 734 | 100 | 4 | 15.033 | 0.298 | 13.650 | 0.308 | $15.66\pm 0.92$ | $0.412\pm 0.002$
NGC6638 V3 | 260.0 | 259.5 | 260.8 | 105 | 584 | 79 | 2 | 15.682 | 0.250 | 13.963 | 0.225 | $9.78\pm 0.34$ | $0.410\pm 0.003$
NGC6638 V15 | 156.0 | 274.8 | 274.7 | 93 | 608 | 73 | 2 | 14.512 | 0.197 | 13.052 | 0.065 | $9.78\pm 0.34$ | $0.376\pm 0.002$
NGC6638 V63 | 213.0 | 214.2 | 215.8 | 122 | 615 | 86 | 3 | 14.232 | 0.299 | 12.827 | 0.450 | $9.78\pm 0.34$ | $0.472\pm 0.004$
NGC6642 V1 | 127.0 | 131.1 | 204.6 | 125 | 621 | 79 | 2 | 14.033 | 0.217 | 12.379 | 0.194 | $8.05\pm 0.20$ | $0.484\pm 0.002$
NGC6712 V7 | 193.0 | 190.2 | 190.4 | 265 | 1093 | 198 | 4 | 12.768 | 0.293 | 11.308 | 0.421 | $7.38\pm 0.24$ | $0.400\pm 0.003$
NGC6760 V3 | 251.0 | 248.4 | 248.7 | 422 | 2152 | 411 | 5 | 15.193 | 0.261 | 13.407 | 0.273 | $8.41\pm 0.43$ | $0.772\pm 0.004$
NGC6760 V4 | 226.0 | 225.9 | 226.0 | 642 | 2770 | 605 | 7 | 15.344 | 0.404 | 13.460 | 0.433 | $8.41\pm 0.43$ | $0.772\pm 0.004$
Palomar4 V1 | 130.0 | 129.7 | 129.7 | 575 | 826 | 451 | 10 | 17.438 | 0.264 | 16.332 | 0.337 | $101.39\pm 2.57$ | $0.124\pm 0.009$
Palomar4 V2 | 150.0 | 150.7 | 150.3 | 557 | 827 | 444 | 8 | 17.111 | 0.228 | 15.950 | 0.288 | $101.39\pm 2.57$ | $0.124\pm 0.009$
Palomar7 V1 | 222.0 | 236.4 | 236.6 | 230 | 688 | 179 | 3 | 16.180 | 0.374 | 14.035 | 0.348 | $4.55\pm 0.25$ | $1.142\pm 0.002$
Palomar7 V3 | 300.0 | 297.9 | 295.2 | 174 | 687 | 147 | 4 | 16.006 | 0.499 | 13.814 | 0.387 | $4.55\pm 0.25$ | $1.146\pm 0.004$
Figure 6: The extinction-corrected PL relations in the $gr$-band for the 14
globular clusters Miras as listed in Table 1. The solid lines are the fitted
PL relations, while the dotted lines represent the $\pm 2.5\sigma$ boundaries.
Note that these PL relations are fitted for magnitudes at maximum light, and
not the mean lights.

By adopting the globular clusters distance $D$ (in kpc) from Baumgardt &
Vasiliev (2021) and the reddenings $E$ from the Bayerstar2019 3D reddening map
(Green et al., 2019, hence the extinction corrections in the $gr$-band are
$3.518E$ and $2.617E$, respectively), we converted the magnitudes at maximum
light listed in Table 1 to absolute magnitudes ($M^{X}_{m}$). The fitted
linear PL relations for our sample of 14 globular clusters Miras, as shown in
Figure 6, are:

$\displaystyle M^{X}_{g}$ $\displaystyle=$ $\displaystyle 4.118[\pm
0.649](\log P-2.3)-2.204[\pm 0.079],$ (2) $\displaystyle M^{X}_{r}$
$\displaystyle=$ $\displaystyle 3.262[\pm 0.562](\log P-2.3)-3.249[\pm
0.074],$ (3)

with a dispersion of 0.577 mag and 0.505 mag, respectively. Iwanek et al.
(2021b) demonstrated that, based on synthetic PL relations at mean light, for
wavelengths shorter than $\sim 0.8\mu$m, slopes of the PL relations are
positive and decrease with wavelengths, at the same time the zero-points (or
intercepts) of the PL relations and the PL dispersions are getting brighter
and smaller, respectively. Our derived $gr$-band PL relations follow these
trends, even though equation (2) and (3) were derived for magnitudes at
maximum light. The positive PL slopes implying that the longer period Miras
are fainter than their short period counterparts.

Miras can be classified into oxygen-rich (O-rich) and carbon-rich (C-rich)
Miras. Using the determined periods $P_{n}$ and $JK$-band photometry, adopted
either from Sloan et al. (2010) or the Two Micron All Sky Survey (2MASS,
Skrutskie et al., 2006), we classified the 14 globular clusters Miras as
O-rich Miras based on the K-nearest neighbor (KNN) algorithm (as done in Ou et
al., 2023). Furthermore, for samples of Miras that extended to $\sim 1000$
days the PL relations were often fitted with either a segmented or a quadratic
relation (for examples, see Ita & Matsunaga, 2011; Yuan et al., 2017, 2018;
Bhardwaj et al., 2019; Iwanek et al., 2021a; Ou & Ngeow, 2022; Sanders, 2023).
In contrast, the longest period in our sample of Miras is $\sim 300$ days
(Palomar7 V3). Hence, strictly speaking the PL relations we derived in
equation (2) & (3) are applicable to O-rich Miras with period less than 300
days.

##  5 Applications to Local Galaxies

Table 2: Observed Properties and the Derived Distance Modulus for Miras in the Local Galaxies. Mira |  $P_{l}$ (days) |  $P_{n}$ (days) | $N_{g}$ | $N_{r}$ | $N_{c}$ | $N_{X}$ | $g_{X}$ | $\sigma_{g}$ | $r_{X}$ | $\sigma_{r}$ |  $E$aaReddening value returned from the Bayerstar2019 3D reddening map (Green et al., 2019). | $\mu_{g}$ | $\mu_{r}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Short-period O-rich Miras
Sextans SDSSJ101234.29-013440.8 | 122 | 120.4 | 369 | 494 | 285 | 6 | 16.929 | 0.115 | 15.887 | 0.073 | $0.072\pm 0.005$ | $19.78\pm 0.67$ | $19.66\pm 0.61$
LeoII CRTSJ111320.6+221116 | 184 | 181.7 | 199 | 454 | 154 | 6 | 19.564 | 0.092 | 18.058 | 0.064 | $0.208\pm 0.002$ | $21.20\pm 0.59$ | $20.90\pm 0.53$
LeoI L8026 | 180 | 198.9 | 88 | 542 | 78 | 2 | 20.285 | 0.050 | 18.571 | 0.093 | $0.248\pm 0.014$ | $21.62\pm 0.58$ | $21.18\pm 0.53$
NGC6822 N12557 | 158 | $\cdots$ | $\cdots$ | 61 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.227 | 0.032 | $0.316\pm 0.002$ | $\cdots$ | $23.12\pm 0.55$
NGC6822 N12790 | 182 | $\cdots$ | $\cdots$ | 42 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.233 | 0.111 | $0.258\pm 0.007$ | $\cdots$ | $23.08\pm 0.54$
NGC6822 N20540 | 223 | $\cdots$ | $\cdots$ | 165 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.200 | 0.043 | $0.276\pm 0.007$ | $\cdots$ | $22.72\pm 0.53$
IC1613 G4237 | 178 | $\cdots$ | $\cdots$ | 29 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.991 | 0.083 | $0.010\pm 0.003$ | $\cdots$ | $24.38\pm 0.54$
Long-period O-rich Miras
NGC6822 N20331 | 314 | $\cdots$ | 39 | 149 | $\cdots$ | $\cdots$ | 19.740 | 0.076 | 18.634 | 0.070 | $0.262\pm 0.002$ | $\cdots$ | $\cdots$
NGC6822 N10184 | 370 | $\cdots$ | $\cdots$ | 119 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 19.524 | 0.162 | $0.258\pm 0.007$ | $\cdots$ | $\cdots$
NGC6822 N30133 | 401 | $\cdots$ | $\cdots$ | 28 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 19.243 | 0.021 | $0.292\pm 0.006$ | $\cdots$ | $\cdots$
NGC6822 N20134 | 402 | $\cdots$ | 4 | 88 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 19.618 | 0.065 | $0.270\pm 0.004$ | $\cdots$ | $\cdots$
NGC6822 N40139 | 545 | $\cdots$ | 44 | 127 | $\cdots$ | $\cdots$ | 20.161 | 0.181 | 18.497 | 0.089 | $0.276\pm 0.007$ | $\cdots$ | $\cdots$
NGC6822 N10198 | 602 | $\cdots$ | 30 | 244 | $\cdots$ | $\cdots$ | 20.400 | 0.069 | 19.462 | 0.018 | $0.206\pm 0.004$ | $\cdots$ | $\cdots$
NGC6822 N30292 | 637 | $\cdots$ | 7 | 203 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 19.470 | 0.161 | $0.302\pm 0.004$ | $\cdots$ | $\cdots$
NGC6822 N10091 | 638 | $\cdots$ | 3 | 39 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 18.918 | 0.028 | $0.206\pm 0.004$ | $\cdots$ | $\cdots$
NGC6822 N20004 | 854 | 693.6 | 180 | 396 | 168 | 2 | 18.033 | 0.327 | 16.351 | 0.363 | $0.276\pm 0.007$ | $\cdots$ | $\cdots$
IC1613 G1016 | 464 | $\cdots$ | 141 | 397 | $\cdots$ | $\cdots$ | 20.021 | 0.092 | 18.825 | 0.088 | $0.000\pm 0.000$ | $\cdots$ | $\cdots$
IC1613 G1017 | 580 | $\cdots$ | 83 | 191 | $\cdots$ | $\cdots$ | 19.605 | 0.106 | 18.508 | 0.100 | $0.000\pm 0.000$ | $\cdots$ | $\cdots$
IC1613 G2035 | 530 | $\cdots$ | 65 | 212 | $\cdots$ | $\cdots$ | 19.832 | 0.081 | 18.730 | 0.059 | $0.000\pm 0.000$ | $\cdots$ | $\cdots$
IC1613 G3011 | 550 | $\cdots$ | 42 | 137 | $\cdots$ | $\cdots$ | 19.606 | 0.041 | 18.581 | 0.057 | $0.000\pm 0.000$ | $\cdots$ | $\cdots$
C-rich Miras
LeoI L1019 | 158 | 160.5 | 54 | 470 | 41 | 1 | 20.634 | $\cdots$ | 18.981 | $\cdots$ | $0.248\pm 0.014$ | $22.36\pm 0.60$ | $21.89\pm 0.54$
LeoI L1077 | 336 | $\cdots$ | $\cdots$ | 55 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.130 | 0.217 | $0.178\pm 0.007$ | $\cdots$ | $22.17\pm 0.62$
NGC6822 N10817 | 214 | $\cdots$ | 100 | 327 | $\cdots$ | $\cdots$ | 20.238 | 0.089 | 19.831 | 0.054 | $0.280\pm 0.005$ | $21.33\pm 0.58$ | $22.25\pm 0.53$
NGC6822 N40590 | 223 | $\cdots$ | $\cdots$ | 78 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.946 | 0.032 | $0.206\pm 0.004$ | $\cdots$ | $23.50\pm 0.52$
NGC6822 N12751 | 231 | $\cdots$ | $\cdots$ | 140 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.389 | 0.098 | $0.258\pm 0.007$ | $\cdots$ | $22.76\pm 0.53$
NGC6822 N11032 | 239 | $\cdots$ | $\cdots$ | 104 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.703 | 0.046 | $0.316\pm 0.002$ | $\cdots$ | $22.87\pm 0.53$
NGC6822 N20578 | 246 | $\cdots$ | $\cdots$ | 61 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.437 | 0.084 | $0.236\pm 0.002$ | $\cdots$ | $22.77\pm 0.54$
NGC6822 N20542 | 255 | $\cdots$ | $\cdots$ | 190 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.251 | 0.034 | $0.276\pm 0.007$ | $\cdots$ | $22.43\pm 0.53$
NGC6822 N30430 | 269 | $\cdots$ | $\cdots$ | 59 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.405 | 0.080 | $0.292\pm 0.006$ | $\cdots$ | $22.47\pm 0.55$
NGC6822 N12208 | 278 | $\cdots$ | 12 | 92 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 19.879 | 0.022 | $0.280\pm 0.005$ | $\cdots$ | $21.93\pm 0.55$
NGC6822 N30583 | 302 | $\cdots$ | 1 | 76 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.954 | 0.074 | $0.254\pm 0.002$ | $\cdots$ | $22.95\pm 0.56$
NGC6822 N40114 | 312 | $\cdots$ | 4 | 35 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.835 | 0.062 | $0.302\pm 0.004$ | $\cdots$ | $22.66\pm 0.57$
NGC6822 N11059 | 319 | $\cdots$ | 1 | 25 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.728 | 0.259 | $0.258\pm 0.007$ | $\cdots$ | $22.64\pm 0.63$
NGC6822 N20657 | 343 | $\cdots$ | $\cdots$ | 25 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.705 | 0.183 | $0.262\pm 0.002$ | $\cdots$ | $22.50\pm 0.61$
NGC6822 N40520 | 432 | $\cdots$ | 3 | 34 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.791 | 0.152 | $0.206\pm 0.004$ | $\cdots$ | $22.41\pm 0.67$
NGC6822 N10753 | 432 | $\cdots$ | $\cdots$ | 115 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.700 | 0.109 | $0.280\pm 0.005$ | $\cdots$ | $22.12\pm 0.66$
NGC6822 N21141 | 456 | $\cdots$ | $\cdots$ | 107 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 19.894 | 0.068 | $0.236\pm 0.002$ | $\cdots$ | $21.36\pm 0.67$
IC1613 G4251 | 263 | $\cdots$ | $\cdots$ | 34 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 21.050 | 0.068 | $0.000\pm 0.000$ | $\cdots$ | $23.91\pm 0.54$
IC1613 G3083 | 280 | $\cdots$ | $\cdots$ | 42 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.689 | 0.229 | $0.000\pm 0.000$ | $\cdots$ | $23.46\pm 0.59$
IC1613 G3144 | 364 | $\cdots$ | $\cdots$ | 31 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 20.671 | 0.175 | $0.000\pm 0.000$ | $\cdots$ | $23.07\pm 0.63$
IC1613 G4183 | 410 | $\cdots$ | $\cdots$ | 17 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 21.040 | 0.066 | $0.000\pm 0.000$ | $\cdots$ | $23.27\pm 0.64$

Ou et al. (2023) demonstrated that ZTF could detect the peak brightness of
known Miras in M33, which is located at a distance of 0.86 Mpc (or
$\mu=24.67\pm 0.07$ mag, de Grijs & Bono, 2014). Therefore, we searched for
local galaxies that are closer than M33, located within the visibility of ZTF,
and contain known Miras detected from NIR observations. We identified five
such galaxies: Sextans (Sakamoto et al., 2012), Leo I (Menzies et al., 2002,
2010), Leo II (Grady et al., 2019), NGC6822 (Whitelock et al., 2013), and
IC1613 (Menzies et al., 2015). Literature distance modulus $\mu$ for Leo I,
Leo II, and Sextans were adopted from Drlica-Wagner et al. (2020) as
$\mu_{\mathrm{Sextans}}=19.7$ mag, $\mu_{\mathrm{Leo~{}I}}=22.0$ mag, and
$\mu_{\mathrm{Leo~{}II}}=21.8$ mag, respectively. In case of NGC6822 and
IC1613, Parada et al. (2023) summarized recent distance measurements to these
two galaxies. We adopted the mid-point of the maximum and minimum $\mu$ listed
in their Table 10 and 11, as $\mu_{\mathrm{NGC6822}}=23.4$ mag and
$\mu_{\mathrm{IC1613}}=24.3$ mag (this value is also same as the recommended
value given in de Grijs & Bono, 2014), respectively.

Figure 7: Examples of $r$-band ZTF light curves for two Miras in NGC6822. The
orange triangles marked the expected epochs of maximum light (based on the
literature period), and the brightest observed data-points tend to be found
near these orange triangles.

Three extragalactic Miras (L2077 in Leo I, N21029 in NGC6822, and
SDSSJ101525.93-020431.8 in Sextans) are known to exhibit a (periodic) long-
term trend, and hence they were excluded from the sample. For the rest of the
Miras in these five local galaxies, we extracted their light curves from the
ZTF DR16 and the partner survey data (in the same way as the globular clusters
Miras, see Section 2). Nevertheless, about half of them either do not have ZTF
light curve data or only contain very few data points in the $g$\- and/or
$r$-band (mostly for the two distant galaxies NGC6822 and IC1613). For the
remaining Miras, we re-classified them into the O-rich and C-rich Miras based
on their $JK$-band photometry published in the aforementioned publications and
the same KNN algorithms applied on the globular clusters Miras.

Based on the quality and the number of data-points on the ZTF light curves, we
analyzed these Miras using two approaches. In case there are enough number of
data-points in the $gr$-band ZTF light curves, we followed the same procedures
as the globular clusters Miras to determine their $P_{n}$, $m_{X}$ and
$\sigma_{X}$. This approach was only applied to five Miras. Otherwise we
adopted the literature periods and only determine the $m_{X}$ and
$\sigma_{X}$, mostly in the $r$-band, by taking the means of the brightest 10%
data-points (that is, following the approach as outlined in Bhardwaj et al.,
2019). This is because for Miras in the distant galaxies, it is possible that
only the portion of the light curves around maximum light could be detected by
ZTF (note that the limiting magnitude for ZTF is $r\sim 20.6$ mag, Bellm et
al., 2019a), as illustrated in Figure 7. We then applied the color-term
corrections on these brightest 10% data-points by adding $C_{m}(g_{X}-r_{X})$
to their calibrated magnitudes, where $C_{m}$ were extracted from the ZTF PSF
catalogs for individual data-points. Assuming the $(g-r)$ colors at maximum
light can be approximated with $(g_{X}-r_{X})$, and fitting the globular
clusters Miras in Table 1 yields:

$\displaystyle(g_{X}-r_{X})\ $ $\displaystyle=$ $\displaystyle\ 0.910[\pm
1.763](\log P-2.3)$ (4) $\displaystyle+1.522[\pm 0.216],$

with a dispersion of $0.299$ mag.555The extinction-corrected version of this
relation is: $(g_{X}-r_{X})_{0}=0.110[\pm 0.846](\log P-2.3)+1.124[\pm
0.107]$, with a dispersion of $0.103$ mag. The measured $P_{n}$ (if
applicable), $m_{X}$, $\sigma_{X}$, and other relevant information are
summarized in Table 2.

|  |
---|---|---
Figure 8: The $gr$-band ZTF light curves for the three short-period O-rich
extragalactic Miras. The symbols are same as in the right panels of Figure 2.

###  5.1 Short-Period O-Rich Miras

Figure 9: Comparison of the $gr$-band absolute magnitudes at maximum light
between globular clusters Miras (open magenta circles) and extragalactic Miras
(in various symbols). For extragalactic Miras, literature $\mu$ (see text for
details) were adopted to convert $m_{X}$ to $M^{X}_{m}$ (after corrected for
extinction). The left and right panels are for the comparisons of O-rich Miras
and C-rich Miras, respectively. The lines are same as in Figure 6, but
extrapolated to a longer period.

There are seven O-rich Miras with period shorter than 300 days (i.e. short
period) detected in all five galaxies. Figure 8 presents three Miras which
have ZTF light curves in both $gr$-band. We calculated the distance moduli for
these seven Miras using our derived PL relations (presented in Section 4),
they are listed in the last two columns of Table 2. The derived distance
moduli for Sextans SDSSJ101234.29-013440.8, NGC6822 N12557 and N12790, as well
as IC1613 G4237 are in good agreement with the literature values. The Mira
CRTSJ111320.6+221116 in Leo II has a smaller distance modulus, especially in
the $r$-band, when compared to the literature $\mu_{\mathrm{Leo~{}II}}=21.8$
mag. Grady et al. (2019) also found a shorter distance ($193\pm 15$ kpc) for
this Mira, using a totally independent dataset and PL relation, with respect
to Leo II ($233\pm 14$ kpc). Therefore, our result is consistent with the
finding of Grady et al. (2019). The $r$-band distance moduli for Leo I L8026
and NGC6822 N20540 were also smaller, nevertheless they are still fall within
$\sim\pm 1.5\sigma$ from the literature values.

By adopting the literature $\mu$, the absolute magnitudes of these seven
short-period extragalactic O-rich Miras were compared to the globular clusters
Miras in the left panels of Figure 9. With the exception of Leo I L8026, all
other short-period O-rich Miras are consistent with globular cluster Miras and
located within the $\pm 2.5\sigma$ boundaries of the $r$-band PL relation.

###  5.2 Long-Period O-Rich Miras

NGC6822 and IC1613 are the only two galaxies in our sample containing O-rich
Miras with period longer than 300 days. The long-period O-rich Miras in
NGC6822 and IC1613, as shown in Whitelock et al. (2013) and Menzies et al.
(2015), respectively, were known to be over-luminous in the $K$-band and the
bolometric magnitude. These over-luminous Miras were presumably undergoing hot
bottom burning (HBB) phase (for examples, see Whitelock et al., 2003; Ita &
Matsunaga, 2011). For the long-period O-rich Miras listed in Table 2, their
$gr$-band absolute magnitudes were also much brighter than the magnitudes
predicted from extrapolating the PL relations to a longer period, as
demonstrated in the left panels of Figure 9. Interestingly, after excluding
the longest-period Mira, these long-period O-rich Miras seems to have a near
constant or mildly period-dependent $gr$-band absolute magnitudes at maximum
light. Since the ZTF light curves are incomplete for these long-period Miras,
and color-term corrections using equation (4) might not be valid for them, the
existence of such a near constant absolute magnitude is not conclusive. The
much deeper and 10 years light curves data collected from LSST can be used to
investigate these long-period Miras further, because both galaxies are located
within the footprint of LSST.

###  5.3 C-Rich Miras

In the era of synoptic sky surveys such as LSST, newly discovered
extragalactic Miras may lack NIR photometry or spectroscopic observations to
be classified to O-rich or C-rich. Hence, the C-rich Miras in Leo I, NGC6822
and IC1613 provide an opportunity to test the distance measurement if a
genuine C-rich Miras discovered from optical surveys was mis-identified as an
O-rich Mira. However, the majority of them only contain incomplete $r$-band
light curves, and suffer the same problems as the long-period O-rich Miras
(see the discussion in sub-section 5.2), the derived distance moduli for these
C-rich Miras, listed in the last two columns of Table 2 using equation (2) and
(3), should be treated with caution.

The $gr$-band distance moduli of the two C-rich Miras in Leo I are in good
agreement with the literature $\mu$ of $\mu_{\mathrm{Leo~{}I}}=22.0$ mag. In
the case of C-rich Miras in both NGC6822 and IC1613, except for a few short-
period Miras, the derived distance moduli were smaller than the literature
values. In other words, they seems to be slightly over-luminous at a given
period, as shown in the right panels of Figure 9. The long-period C-rich Miras
are also tend to be over-luminous, similar to the cases of long-period O-rich
Miras. The right panels of Figure 9 suggested the C-rich Miras seems to have a
flat PL relation in the $r$-band.

##  6 Discussion and Conclusions

In this work, we derived the $gr$-band PL relations for the O-rich and short-
period ($<300$ days) Miras in the globular clusters using the light-curve data
collected from ZTF, as these globular clusters possess homogeneous distances
compiled in Baumgardt & Vasiliev (2021). We focus on the maximum light when
deriving the PL relations because Miras are known to exhibit a smaller PL
dispersion at maximum light when compared to their mean-light counterparts.
Furthermore, the $(g-r)$ colors tend to be bluest at maximum light, implying
the color-term corrections, $C_{m}(g-r)$, would be smaller when compared to
other pulsation phases such as at mean light. Finally, due to the large-
amplitude nature of Miras, it is possible that only the portions of the light
curves around the maximum light could be detected for Miras located in distant
galaxies. In this scenario, measurements at maximum light can still be used to
derive the distance moduli to the host galaxies, as we demonstrated in Section
5 for the Miras in NGC6822 and IC1613. Given that the limiting magnitude for
ZTF is $r\sim 20.6$ mag (Bellm et al., 2019a), the maximum light for a Mira
with $\log P=2.3$, based on equation (3), could be detected at a distance
modulus of $\mu\sim 23.9$ mag, which is within $\pm 0.5$ mag from
$\mu_{\mathrm{NGC6822}}=23.4$ mag and $\mu_{\mathrm{IC1613}}=24.3$ mag. For
LSST, the single-epoch limiting magnitude in the $r$-band is $\sim 24.7$ mag,
implying a $\log P=2.3$ Mira could be detected up to $\mu\sim 28$ mag at
maximum light, corresponding to a distance of $\sim 4$ Mpc.

In Figure 10, we presented the $gr$-band PL relations at maximum light for all
of the Miras investigated in this work. In the $r$-band, which has more Miras
than the $g$-band, both of the short-period O-rich and C-rich Miras seem to
occupy the same region in the PL plane, suggesting our derived $gr$-band PL
relations could also be used for the short-period C-rich Miras (also, see the
discussion in Section 5.3). However, as periods become longer, the long-period
C-rich Miras appear to “connect” to the long-period HBB O-rich Miras.
Altogether, the short- and long-period Miras formed a quadratic or segmented
relation, hence the linear PL relations should only be applied to the short-
period Miras. We suggest using short-period extragalactic Miras detected with
LSST for distance measurements not only because they follow a linear relation,
but also their periods will be better constraint from the 10 years
observations.

Figure 10: The PL relations at maximum lights for all of the Miras listed in
Table 1 and 2. The symbols are same as in Figure 9. However, for a better
visualization, open and filled symbols were used represent all of the O-rich
and C-rich Miras, respectively.

We are thankful for funding from the National Science and Technology Council
(Taiwan) under the contracts 107-2119-M-008-014-MY2, 107-2119-M-008-012,
108-2628-M-007-005-RSP and 109-2112-M-008-014-MY3. Based on observations
obtained with the Samuel Oschin Telescope 48-inch Telescope at the Palomar
Observatory as part of the Zwicky Transient Facility project. ZTF is supported
by the National Science Foundation under Grants No. AST-1440341 and
AST-2034437 and a collaboration including current partners Caltech, IPAC, the
Weizmann Institute of Science, the Oskar Klein Center at Stockholm University,
the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt
University, the TANGO Consortium of Taiwan, the University of Wisconsin at
Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories,
IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University
and former partners the University of Washington, Los Alamos National
Laboratories, and Lawrence Berkeley National Laboratories. Operations are
conducted by COO, IPAC, and UW. This publication makes use of data products
from the Two Micron All Sky Survey, which is a joint project of the University
of Massachusetts and the Infrared Processing and Analysis Center/California
Institute of Technology, funded by the National Aeronautics and Space
Administration and the National Science Foundation. This research has made use
of the SIMBAD database and the VizieR catalogue access tool, operated at CDS,
Strasbourg, France. This research made use of
Astropy,666http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018, 2022). PO:1.2m

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aainstitutetext: Bogoliubov Laboratory of Theoretical Physics JINR, 141980
Dubna, Russiabbinstitutetext: Institute of Modern Physics, Chinese Academy of
Sciences, 730000 Lanzhou, P.R. Chinaccinstitutetext: Southern Center for
Nuclear Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of
Sciences, 516000 Huizhou, P.R. China

# Inverse Radon transforms: analytical and Tikhonov-like regularizations of
inversion

I. V. Anikin b, c Xurong Chen<EMAIL_ADDRESS>
###### Abstract

We study the influence of analytical regularization used in the generalized
function (distribution) space to the Tikhonov regularization procedure
utilized in the different versions of Moore-Penrose’s inversion. By
introducing a new analytical term to the Tikhonov regularization of Moore-
Penrose’s inversion procedure, we derive new optimization conditions that
extend the Tikhonov regularization framework and influence the fitting
parameter. This enhancement yields a more robust and accurate reconstruction
of physical quantities, demonstrating its potential impact on various studies.
We illustrate the significance of new term through schematic examples of
physical applications, highlighting its relevance to diverse fields. Our
findings provide a valuable tool for improving inversion methods and their
applications in physics and beyond.

##  1 Introduction

As well-known, the mathematical basis of reconstructive (computerized)
tomography is formed by the direct and inverse Radon transforms Deans . The
reconstructive tomography gives us a possibility to explore the internal
structure of a given object without breaking it. However, the inverse Radon
transformation is more difficult mathematical task compared to the direct
Radon transformation 111For the Radon transforms, the inverse operators have
different representations depending on the space dimension, even or odd
dimension.:

$\displaystyle{\cal R}[f]=G\quad\Longrightarrow\quad f={\cal R}^{-1}G$ (1)

because we are forced to deal with the inverse operator that demands some
suitable regularization.

Frequently, in the most practical cases, the inverse transforms suffer from
the ill-posedness. According Hadamard’s criteria, some mathematical problems
are called to be well-posed ones provided they fulfil the following
conditions: (a) existence (i.e. a solution of the problem exists) (b)
uniqueness (i.e. the given solution is unique) and (c) stability (i.e. the
given solution depends on the data only-continuously) Bellmann ; Kazakovtsev ;
Baguer ; Kabri . In opposite cases, we deal with the ill-posedness.

Moreover, in order to fully-reconstruct the internal distribution we have to
implement the infinite number of Radon transforms 222This statement is a
consequence of the indeterminacy theorem: a given origin function $f$ with the
compact support is uniquely determined by any infinite set of its
projections.. Of course, “infinite” is impossible from the practical point of
view, therefore the infinite number should be replaced by a reasonable number
which is the other aspect of difficulties related to the Radon transforms.

To invert the Radon transform, in both even and/or odd dimension of space,
there are the standard methods to do that (see for example Deans ; GGV ).
Within the standard methods, in order to derive the inverse Radon transforms
we have to integrate with the symmetric angular measure:

$\displaystyle{\cal R}^{-1}\sim\int_{|\xi|=1}d\xi(\varphi)\,....\,,$ (2)

where $\xi(\varphi)$ is defined by the corresponding angle $\varphi$ and
$\varphi\in[0,\,2\pi]$. Here, the explicit integrand that is not important at
the moment is omitted. Notice that the integration measure of (2) is mostly
associated with the emission imaging where the radiation source is located
inside the scanned object. It cannot, however, be realized or might be
realized with some difficulties in many cases of practices.

The most practically attractive kind of computerized tomography is provided by
the transmission imaging where the radiation source is out of the scanned
object. In this case, as shown in Anikin:2019oes , any well-localized original
internal function (distribution), which describes the scanned object, must
lead to the angular restrictions on the integration measure of inverse Radon
transforms. In other words, the angular measure of integration for ${\cal
R}^{-1}$ receives the angular constraints, $\varphi\in[-\pi/2,\,\pi/2]$. As a
result, we deal with the additional new term $f_{A}$ contributing the inverse
Radon transforms that is absent in the standard method.

Previously, in Anikin:2019oes , the practical application of new term $f_{A}$
in QFT has been demonstrated in the case where the transverse momenta of
partons inside hadrons are very essential. In the present paper, we show a
role of new term $f_{A}$ in the Moore-Penrose inversion procedure where the
Tikhonov-like regularization has been utilized.

##  2 The inverse Radon transformations and analytical regularization

In this section, we consider the analytical regularization of the inverse
Radon transform which is inspired by the method used for the singular
generalized (distribution) functions. For the sake of simplicity, we begin
with the inverse Radon transforms Anikin:2019oes  in two-dimensional space. As
well-known, to derive the inverse Radon transform we first have to invert the
Fourier transform given by

$\displaystyle\mathcal{F}[f](\vec{\bf
q})=\int_{-\infty}^{+\infty}d^{2}\vec{\bf x}\,e^{-i\langle\vec{\bf q},\vec{\bf
x}\rangle}\,f(\vec{\bf x}).$ (3)

Generally speaking, both $\mathcal{F}[f](\vec{\bf q})$ and $f(\vec{\bf x})$
can be the complex functions, i.e. $\\{\mathcal{F}[f](\vec{\bf
q}),\,f(\vec{\bf x})\\}\in\mathds{C}$.

It is worth to notice that the Fourier (direct or inverse) transform is always
a good starting point for many known transformations. Indeed, with the help of
the corresponding replacements of variables, the Fourier transform gives the
Laplace transforms (due to the replacement $ix=\omega$ in the Fourier
transforms) or the Borel transforms (due to the replacement $\omega=1/y$ in
the Laplace transforms) and so on.

Thus, the inversion procedure yields

$\displaystyle f(\vec{\bf x})=\int_{-\infty}^{+\infty}d^{2}\vec{\bf
q}\,e^{+i\langle\vec{\bf q},\vec{\bf x}\rangle}\,\mathcal{F}[f](\vec{\bf
q})\Big{|}_{\vec{\bf q}=\lambda\vec{\bf
n}_{\varphi}}=\int_{0}^{+\infty}d\lambda\lambda\int_{0}^{2\pi}d\varphi\,e^{+i\lambda\langle\vec{\bf
n}_{\varphi},\vec{\bf x}\rangle}\,\mathcal{F}[f](\lambda,\varphi)$ (4)

where the polar coordinates have been used. We now remind the Fourier slice
theorem that merely states

$\displaystyle\mathcal{F}[f](\lambda,\varphi)=\int_{-\infty}^{+\infty}(d\tau)\,e^{-i\lambda\tau}\,\mathcal{R}[f](\tau,\varphi),$
(5)

where

$\displaystyle\mathcal{R}[f](\tau,\varphi)=\int_{-\infty}^{+\infty}d^{2}\vec{\bf
x}\,f(\vec{\bf x})\delta\left(\tau-\langle\vec{\bf n}_{\varphi},\vec{\bf
x}\rangle\right)$ (6)

defines the direct Radon transformation of $f(\vec{\bf x})$. Using (5), after
simple algebra (4) can be presented in a form of

$\displaystyle f(\vec{\bf
x})=\int_{0}^{2\pi}d\varphi\,\int_{-\infty}^{+\infty}(d\eta)\,\mathcal{R}[f](\eta+\langle\vec{\bf
n}_{\varphi},\vec{\bf
x}\rangle,\varphi)\int_{0}^{+\infty}d\lambda\lambda\,e^{-i\lambda\eta}.$ (7)

From this, we can readily observe that the integration over $\lambda$ suffers
from the singularity at the infinite integral limit. In other words, this
integral representation demands some regularization. It is not something
unexpected in the generalized function theory where the regularization is
introduced by $\eta\to\eta-i\epsilon$. This is the so-called analytical
regularization. Given that, for the $\lambda$-integration, one gets

$\displaystyle
i\frac{\partial}{\partial\eta}\int_{0}^{+\infty}d\lambda\,e^{-i\lambda(\eta-i\epsilon)}\equiv
2\pi\frac{\partial}{\partial\eta}\delta_{-}(\eta)=-\frac{\mathcal{P}}{\eta^{2}}+i\pi\frac{\partial}{\partial\eta}\delta(\eta),$
(8)

where $\lambda$ has been traded for the derivative over $\eta$ acting on the
exponential function.

We finally derive for the inverse Radon transform in a form of

$\displaystyle f(\vec{\bf x})=f_{S}(\vec{\bf x})+f_{A}(\vec{\bf x}),$ (9)

where

$\displaystyle f_{S}(\vec{\bf
x})=-\int_{-\infty}^{+\infty}(d\eta)\,\frac{\mathcal{P}}{\eta^{2}}\,\int_{0}^{2\pi}d\varphi\,\mathcal{R}[f](\eta+\langle\vec{\bf
n}_{\varphi},\vec{\bf x}\rangle,\varphi)$ (10)

is the standard contribution, while

$\displaystyle f_{A}(\vec{\bf
x})=-i\pi\int_{-\infty}^{+\infty}(d\eta)\,\delta(\eta)\,\int_{0}^{2\pi}d\varphi\Big{[}\frac{\partial}{\partial\eta}\mathcal{R}[f](\eta+\langle\vec{\bf
n}_{\varphi},\vec{\bf x}\rangle,\varphi)\Big{]}$ (11)

corresponds to the new term found in Anikin:2019oes  provided that the angular
integration is limited by the interval $[-\pi/2,\,\pi/2]$ instead of the full
interval $[0,\,2\pi]$. That is, it can be proven that

$\displaystyle\boxed{f_{A}(\vec{\bf x})\not=0\hskip 8.5359pt\text{iff}\quad
d\varphi\Rightarrow
d\varphi\,\Theta\big{(}\varphi\in[-\pi/2,\,\pi/2]\big{)}\hskip
5.69046pt\text{in (\ref{Inv-F-t-3-b})}.}$ (12)

As explained in Anikin:2019oes , the mentioned restriction on the angular
integration can be traced from the fact that the compact origin function $f$
has led to the angular restriction for its Fourier image. In its turn, the
angular dependences of Fourier and Radon transforms are equivalent to each
other, see (5).

In (11) and (12), these equations actually gives the integral representation
of $f_{A}$, the derivative over $\eta$ can be dragged to $\mathcal{R}[f]$
without the surface terms. It follows from the theorem asserting that the
direct Radon transform of the function with the localized (or restricted)
support is also a compact function Deans .

###  2.1 The operator form of analytical regularization

For our further purposes, let us rewrite the analytical regularization
procedure in a form of operators. At the first, we introduce the operator of
direct Radon transforms as (c.f. (6))

$\displaystyle
G=\mathcal{R}[f]\quad\text{with}\quad\mathcal{R}[...]=\int_{-\infty}^{+\infty}d^{2}\vec{\bf
x}\,\delta\left(\tau-\langle\vec{\bf n}_{\varphi},\vec{\bf
x}\rangle\right)[....]$ (13)

and, then, the inverse and regularized inverse Radon transforms take the
operator form of (c.f. (9)-(12))

$\displaystyle f=\mathcal{R}^{-1}[G]\quad\Longrightarrow\quad\lim_{\epsilon\to
0}f_{\epsilon}\quad\text{where}\quad$ $\displaystyle
f_{\epsilon}=\mathcal{R}^{-1}_{\epsilon}[G]\quad\text{with}\quad$
$\displaystyle\mathcal{R}^{-1}_{\epsilon}[...]=-\int_{-\infty}^{+\infty}(d\eta)\,\frac{\mathcal{P}}{\eta^{2}}\,\int_{-\pi/2}^{\pi/2}d\varphi\,[...]-i\pi\int_{-\infty}^{+\infty}(d\eta)\,\delta(\eta)\,\int_{-\pi/2}^{\pi/2}d\varphi\,[...].$
(14)

In (13)-(2.1) the argument dependences are not shown explicitly.

###  2.2 The dual Radon transforms and analytical regularization

It is also instructive to introduce the dual Radon transforms defined as

$\displaystyle\mathcal{R}^{\ast}[G]\equiv\mathcal{R}^{\ast}[G](\vec{\bf
x})=\int_{-\infty}^{+\infty}d\tau\int_{0}^{2\pi}d\varphi\,G(\tau,\varphi)\delta\left(\tau-\langle\vec{\bf
n}_{\varphi},\vec{\bf
x}\rangle\right)=\int_{0}^{2\pi}d\varphi\,G(\langle\vec{\bf
n}_{\varphi},\vec{\bf x}\rangle,\varphi).$ (15)

Making used the fact that $G(\tau,\varphi)$ is being
$\mathcal{R}[f](\tau,\varphi)$, we can write the following

$\displaystyle\mathcal{R}^{\ast}\mathcal{R}[f](\vec{\bf
x})=\int_{0}^{2\pi}d\varphi\,\mathcal{R}[f](\langle\vec{\bf
n}_{\varphi},\vec{\bf x}\rangle,\varphi).$ (16)

In order to see the other realization of analytical regularization applied to
the Radon transform, we now consider the inverse Fourier transform of
$1/|\vec{\bf q}|$. We have

$\displaystyle{\cal F}^{-1}\Big{[}\frac{1}{|\vec{\bf
q}|}\Big{]}\Big{|}_{\vec{\bf q}=\lambda\vec{\bf
n}_{\varphi}}=\int_{0}^{2\pi}d\varphi\int_{0}^{+\infty}d\lambda\,e^{+i\lambda\langle\vec{\bf
n}_{\varphi},\vec{\bf x}\rangle}.$ (17)

As in (7), the integration over $\lambda$ requires the analytical
regularization which finally leads to

$\displaystyle{\cal
F}^{-1}_{\epsilon}\Big{[}\frac{1}{\lambda}\Big{]}=\int_{0}^{2\pi}d\varphi\,\frac{1}{\langle\vec{\bf
n}_{\varphi},\vec{\bf
x}\rangle+i\epsilon}=\int_{0}^{2\pi}d\varphi\int_{-\infty}^{\infty}d\tau\,\frac{1}{\tau+i\epsilon}\delta(\tau-\langle\vec{\bf
n}_{\varphi},\vec{\bf
x}\rangle)\equiv{\mathcal{R}}^{*}\Big{[}\frac{1}{\tau+i\epsilon}\Big{]}.$ (18)

On the other hand, we can derive that (here, the argument dependence has been
shown explicitly)

$\displaystyle{\cal
F}\Big{[}{\mathcal{R}}^{\ast}{\mathcal{R}}[G]\Big{]}(\lambda\vec{\bf
n}_{\varphi})=\frac{1}{\lambda}{\cal F}[G](\lambda\vec{\bf n}_{\varphi}).$
(19)

If we assume that $G(\vec{\bf x})=e^{+i\langle\vec{\bf Q},\vec{\bf x}\rangle}$
(this is merely a plane wave function), then one can easily obtain that

$\displaystyle{\cal
F}\Big{[}{\mathcal{R}}^{\ast}{\mathcal{R}}[e^{+i\langle\vec{\bf Q},\vec{\bf
x}\rangle}]\Big{]}(\vec{\bf q})=\frac{1}{|\vec{\bf
Q}|}\delta^{(2)}\big{(}\vec{\bf q}-\vec{\bf Q}\big{)}.$ (20)

or

$\displaystyle{\mathcal{R}}^{\ast}{\mathcal{R}}[e^{+i\langle\vec{\bf
Q},\vec{\bf x}\rangle}]=\frac{1}{|\vec{\bf Q}|}e^{+i\langle\vec{\bf
Q},\vec{\bf x}\rangle}.$ (21)

Hence, from (21) one can conclude that $1/|\vec{\bf Q}|$ is an eigenvalue of
operator ${\mathcal{R}}^{\ast}{\mathcal{R}}$ and the plane wave is its
eigenfunction. In its turn, it means that the singular value of
${\mathcal{R}}^{-1}$ is going to infinity if $|\vec{\bf Q}|\to\infty$, i.e.
(see Appendix A for the definition of singular values $\sigma_{A}$ of
matrix/operator $A$)

$\displaystyle\sigma_{{\mathcal{R}}^{-1}}=\sqrt{|\vec{\bf Q}|}\to\infty.$ (22)

As a result, the inverse Radon operator is an unbounded that leads to the
amplification of noises.

Taking into account (18) and (22), we can infer that in order to get the
bounded support for the inverse Radon transforms we have to make the definite
regularization applied to the Radon parameter as $\tau\to\tau+i\epsilon$. That
is, we deal with the following regularized Radon transforms

$\displaystyle\mathcal{R}_{\epsilon}[f](\tau,\varphi)\equiv\mathcal{R}[f](\tau+i\epsilon,\varphi)=\int_{-\infty}^{+\infty}d^{2}\vec{\bf
x}\,f(\vec{\bf x})\delta\left(\tau+i\epsilon-\langle\vec{\bf
n}_{\varphi},\vec{\bf x}\rangle\right)=G_{\epsilon}.$ (23)

Notice that the singular character of the nullified radial Radon parameter can
be understood as the degeneration of the nullified radial parameter in the
polar system of coordinates. Indeed, let us consider a very illustrative
example of $\mathbb{R}^{2}$ where two vectors $X$ and $Y$ emanate from the
same starting point $O$. In the frame of polar (or spherical) system, we have
$X(r_{X},\phi_{X})\equiv(r_{X}\cos\phi_{X},r_{X}\sin\phi_{X})$ and the similar
representation can be written for $Y$. If the radius vectors of both $X$ and
$Y$ are not equal to zero, even if they are infinitesimal ones, it is possible
to distinguish these two vectors. However, if $r_{X}=r_{Y}=0$, the starting
point $O$ loses somehow information on the differences of vectors $X$ and $Y$
due to the condition $O=(|\vec{\bf 0}|\,\cos\phi_{X},|\vec{\bf
0}|\,\sin\phi_{X})=(|\vec{\bf 0}|\,\cos\phi_{Y},|\vec{\bf 0}|\,\sin\phi_{Y})$.
It leads to the degeneration of the starting point $O$ (or $\tau=0$) in the
Radon transforms (see also Anikin:2021oht ).

Thus, based on (18)-(22) we have demonstrated that the unbounded support of
the Radon transforms is nothing but the singularity (degeneration) of the case
where the radial Radon parameter goes to zero.

##  3 The extended Tikhonov regularization of inversion

Building on the analytical regularization of the inverse Radon operator
presented in the preceding section, we now explore alternative approaches to
address the challenges of inverting singular operators. While our
regularization is inspired by the theory of generalized functions, the other
methods have been developed to deal with this problem. One of such approaches
is Tikhonov’s regularization Kazakovtsev ; Baguer ; Kabri  which implements
Moore-Penrose’s inversion procedure MP  through a different realization. This
indirect inversion method offers a distinct perspective on the problem,
underscoring the need for a comprehensive understanding of the various
strategies which are available for addressing the singular operator inversion.

To demonstrate the method, we consider the operator representation of the
direct Radon transforms, see (13). Let us now suppose that the system given by
(13) is not compatible one (this is rather a typical situation for the ill-
posed inversion), i.e. instead of (13) we have

$\displaystyle G\approx{\cal R}[f].$ (24)

According the normal equation theorem, see Appendix A, we infer that the best
approximation for the solution of this system, first, can be found by
minimization of the norm

$\displaystyle\min_{f}\big{\\{}\|{\cal R}[f]-G\|\big{\\}}$ (25)

and, then, it is given by a solution of the system (if ${\cal R}^{\ast}\sim
A^{T}$ and ${\cal R}\sim A$, see Appendix A)

$\displaystyle{\cal R}^{\ast}\,{\cal R}[f]={\cal
R}^{\ast}[G]\quad\Longrightarrow\quad f=\frac{{\cal R}^{\ast}}{{\cal
R}^{\ast}\,{\cal R}}[G].$ (26)

Using (21), one can immediately see that the operator $[{\cal R}^{\ast}\,{\cal
R}]^{-1}$ demands some regularization for the well-definitness (boundedness)
of solution $f$. We introduce the regularized operator as

$\displaystyle{\cal R}^{\ast}\,{\cal R}+\Lambda\,{\cal
I}\quad\Longrightarrow\quad f_{reg.}=\frac{{\cal R}^{\ast}}{{\cal
R}^{\ast}\,{\cal R}+\Lambda\,{\cal I}}[G].$ (27)

This kind of regularizations implies that the norm of (25), that should be
minimized, has to be modified too. Thus, the optimitization procedure should
be applied for the following modified norm (in the literature, it is usually
called as Tikhonov’s regularization):

$\displaystyle\min_{f}\big{\\{}\|{\cal
R}[f]-G\|^{2}+\Lambda\,\|f\|^{2}\big{\\}},$ (28)

where $\Lambda$ should be chosen for the best approximation.

In (13), the analytical regularization of the inverse Radon operator
(transform) has been introduced. In order to get the matched correspondence
between the Tikhonov regularization and our analytical regularization, we
propose to minimize the norm as

$\displaystyle\boxed{\min_{f}\big{\\{}\|{\cal
R}[f]-G\|^{2}+\Lambda\,\|f_{\epsilon}\|^{2}\big{\\}}.}$ (29)

Alternatively, the optimization procedure can be applied to

$\displaystyle\min_{f}\big{\\{}\|{\cal
R}[f]-G_{\epsilon}\|^{2}+\Lambda\,\|f\|^{2}\big{\\}}.$ (30)

The key findings of the present paper are encapsulated in the conditions
presented by (29) and (30). A close examination of these conditions reveals
that the additional term $f_{\epsilon}$ (or $G_{\epsilon}$) stemmed from the
analytically regularized inversion operator plays a crucial role in the
Tikhonov-type regularization. This term has a significant impact on the
fitting parameter $\Lambda$.

##  4 The potential applications

In this section, we explore the potential physical applications of the newly
derived terms. Notably, the direct Radon transform has been extensively
utilized in the medical imaging, in particular, in emission and transmission
reconstructed tomography. The logarithmic ratio of the input and output
intensities, obtained after an X-ray or proton beam passes through the scanned
object, is equivalent to the direct Radon transform of the damping function
$f_{d}(x;\rho,Z)\equiv f_{d}(x)$ 333Here, $\rho$ and $Z$ imply the material
density and the atomic number, respectively., i.e.

$\displaystyle-\log\Big{[}\frac{I}{I_{0}}\Big{]}=\int_{L(\tau,\varphi)}d\sigma\,f_{d}\big{(}x(\sigma);\rho,Z\big{)}\equiv{\cal
R}[f_{d}](\tau,\varphi),$ (31)

where $L(\tau,\varphi)$ implies the line parametrization with the radial and
angular parameters $\tau$ and $\varphi$, respectively. This representation
gives the so-called linear realization where the energy $E$ of beam is fixed.
The l.h.s. of (31) relates to the observable value measured by apparatus.

To circumvent the degeneracy point ($\tau=0$) of the Radon transform, we
employ a suitable analytical regularization of the direct Radon transform,
${\cal R}[f_{d}]\to{\cal R}_{\epsilon}[f_{d}]$ as described by (23). This
regularization gives an additional contribution to (31) due to the imaginary
part of ${\cal R}_{\epsilon}[f_{d}]$.

The case of fixed X-ray energy is usually not a very good for the practical
applications. As a rule, the energy has some distribution $w(E)$ in the known
interval. It means that we practically deal with the following representation

$\displaystyle\frac{I}{I_{0}}=\int
d\mu(E,w)\,\exp\Big{\\{}-\int_{L(\tau,\varphi)}d\sigma\,f_{d}\big{(}x(\sigma),E)\Big{\\}}\Rightarrow\int
d\mu(E,w)\,e^{-{\cal R}_{\epsilon}[f_{d}\big{(}x(\sigma),E)]},$ (32)

where the integration measure is defined with the corresponding weight
function $w$. This is already the non-linear realization due to the
integration measure is presented.

A striking resemblance is observed between the representation of $I/I_{0}$
given by (32) and the case of essential transverse momenta in the
corresponding parton distribution functions considered in QFT, see
Anikin:2019oes , where the imaginary part of ${\cal R}_{\epsilon}^{-1}$ (or
${\cal R}_{\epsilon}$) generates an extremely important contribution. Indeed,
the most natural correspondence can be established by mapping the line
position $x(\sigma)$ to the parton longitudinal fraction $y$ and the X-ray (or
beam) energy $E$ to the parton transverse momenta $k_{\perp}$. This
correspondence enables a clearer understanding of the other sources of
imaginary parts and highlights the potential for further connections between
these seemingly disparate frameworks.

Last but not least, the extension of Tikhonov’s regularization defined by (29)
can be applied to the iterative algorithms used for the reconstruction from
projections. It is very close to the inversion problems considered in Section
3. In this case, the direct Radon operator $\mathcal{R}$ (and its the
regularized version $\mathcal{R}_{\epsilon}$) has to be presented in the
matrix form. To do that, let us rewrite the integral representation of direct
Radon transforms as

$\displaystyle\mathcal{R}[f](\tau,\varphi)=\int_{L}d\sigma\,f\big{(}\tau\,\vec{\bf
n}(\varphi)+\sigma\,\vec{\bf n}_{\perp}(\varphi)\big{)},$ (33)

where $\vec{\bf n}(\varphi)=(\cos\varphi,\sin\varphi)$ and $\vec{\bf
n}_{\perp}(\varphi)=(-\sin\varphi,\cos\varphi)$.

Now, we are going over to the digital picture of the origin function $f$. In
other words, the function $f$ has been represented by its digitized version
(in pixels). This kind of discrete approximations leads to the matrix forms
given by (c.f. (33))

$\displaystyle\mathcal{R}_{i,\,j}[f]\equiv\mathcal{R}[f](\tau_{i},\varphi_{j})=\sum_{M,\,K=1}^{n}\Delta\sigma_{M\,K}\,f_{M\,K\,;\,i,\,j}\,,$
(34)

where summation goes over the pixel squares presented in the matrix form and
$\tau_{i}\,,\varphi_{j}$ are temporally fixed but they finally vary within the
pixel picture. We also assume that $1<i<n$ and $1<j<n^{\prime}$. Notice that,
alternatively, (34) can be written as

$\displaystyle{\bf R}=\hat{\bf F}\,{\bf S},$ (35)

where ${\bf R}$ is a column matrix with elements marked by $n\,n^{\prime}$,
$\hat{\bf F}$ defines the $n\,n^{\prime}\times n^{2}$ matrix related to
$f_{M\,K\,;\,i,\,j}$ and ${\bf S}$ is a column matrix with $n^{2}$ number of
elements and associates with $\Delta\sigma_{M\,K}$. Given (35), one can apply
the extended Tikhonov regularization, see (29), to reconstruct the (digital)
structure of the origin function $f$.

##  5 Conclusions

In this paper, we have demonstrated the significant role of the new term
$f_{A}$ (or $f_{\epsilon}$) in the Moore-Penrose inversion procedure which has
been regularized using the Tikhonov-like approach. We have introduced the
novel conditions, see (29) and (30), on the optimization procedure extending
the scope of the Tikhonov regularization and influencing the fitting parameter
$\Lambda$ in the inversion process. Our results have the far-reaching
implications for various applications where the additional term in
$f_{\epsilon}$ (or in $G_{\epsilon}$), see (2.1), arising from the analytical
regularization, can play a crucial role in different studies.

## Acknowledgements

We thank G. I. Andreev, Sh. Z. Sharipov, D. Strozik-Kotlorz, L. Szymanowski
and Jian-Hui Zhang for useful and illuminating discussions. Also, I.V.A.
thanks his colleagues from the Chinese University of Hong Kong for a very warm
hospitality. The work has been supported in part by PIFI 2024PVA0110 Program.

## Appendix

##  Appendix A Least Squares Optimization and Singular Value Decomposition

In this appendix, for the pedagogical reason we remind the basic material of
linear algebra that is needed for our study, see for example Bellmann . Let us
begin with an operator $A:H_{1}\to H_{2}$ where $H_{i}$ can be assumed to be
the Hilbert space. We want to find $x\in H_{1}$ that satisfies the equation
$Ax=y$. As well-known, a solution $x$ of this equation can be found uniquely
if and only if $Ker(A)={0}$ where $Ker(A):\\{x\in H_{1}\,|\,Ax=0\\}$.

Now let $A$ be an $m\times n$ matrix with $m>n$ and $Rank(A)=n$. The least
squares approximation (LSA) of the system $Ax\approx b$ is a vector $x$ that
minimizes the norm given by $\|\,Ax-b\,\|$ where the norm is defined as
$\|\,Y\,\|_{p}=(\sum_{i=1}^{n}|\,Y\,|^{p})^{1/p}$ for $p\in\mathbb{Z}$. An
alternative definition of LSA can be formulated as follows: LSA of $Ax\approx
b$ is a solution of the system given by $A^{T}Ax=A^{T}b$. This is a theorem on
the normal equation (or the normal equation theorem).

One of the fundamental method of matrix representation widely used in machine
learning and data science is the singular value decomposition (SVD). This
decomposition allows to reduce the dimensionality and the noise influence
together with the realization of data vision. To explain the main idea of SVD,
we consider again the above-mentioned matrix $A$. Then, we construct the
combinations $A\,A^{T}$ and $A^{T}\,A$ which are now symmetric matrices and,
therefore, both matrices can be orthogonally diagonalized, i.e.
$A\,A^{T}=U\,D_{1}\,U^{T}$ and $A^{T}\,A=V\,D_{2}\,V^{T}$. These
decompositions give the singular value decomposition in a form of $A=U\Sigma
V^{T}$ where $\Sigma$ is a diagonal $m\times n$ matrix containing the singular
values which are, by definition, equal to the square roots of the eigenvalues
of $D_{1}$ and $D_{2}$.

Notice that all eigenvalues $\lambda$ of $A\,A^{T}$ and $A^{T}\,A$ (these
matrices have the same set of eigenvalues) are positive ones, i.e.
$\lambda\geq 0$. Moreover, if $p$ is an eigenvector of $A\,A^{T}$ with
$|p|=\langle p,p\rangle=1$, then $q=A^{T}p/\sigma$ is an eigenvector of
$A^{T}\,A$ where $\sigma=\sqrt{\lambda}$ defines the set of singular values of
$A$. One can easily see that it takes place the converse relation given by
$p=A\,q/\sigma$.

## References

  * (1) S.R. Deans, “The Radon Transform and Some of Its Applications,” Wiley 299 p (1983).
  * (2) R. Bellmann, “Introduction to Matrix Analysis,” The RAND Corporation. 403 p (1997).
  * (3) V. Stanojevic, L. Kazakovtsev, P.S. Stanimirovic et al, Algorithms 15, 348 (2022)
doi:10.3390/a15100348

  * (4) D. O. Baguer, J. Leuschner and M. Schmidt, Inverse Problems 36, 094004 (2020)
doi:10.1088/1361-6420/aba415

  * (5) S. Kabri, A. Auras, D. Riccio, H. Bauermeister et al, Communications on Applied Mathematics and Computation (2024)
doi:10.1007/s42967-023-00333-2

  * (6) I.M. Gelfand, M.I. Graev, N.Ya. Vilenkin, “Generalized Functions, Volume 5: Integral Geometry and Representation Theory,” AMS Chelsea Publishing: An Imprint of the American Mathematical Society, 449 pp (1966).
  * (7) I. V. Anikin and L. Szymanowski, Phys. Rev. D 100, no.9, 094034 (2019) doi:10.1103/PhysRevD.100.094034 [arXiv:1909.00017 [hep-ph]].
  * (8) I. V. Anikin and A. S. Zhevlakov, J. Exp. Theor. Phys. 135, no.1, 73-80 (2022) doi:10.1134/S1063776122070081 [arXiv:2105.02719 [hep-ph]].
  * (9) E. H. Moore, Mem. Amer. Phil Soc. 1, 197 (1935);
R. Penrose, Proc. Cambridge Phil. Soc. 51 406 (1953)


# On Completely Invariant Julia Sets of Transcendental Semigroups

Bishnu Hari Subedi Central Department of Mathematics
Institute of Science and Technology
Tribhuvan University
Kirtipur, Kathmandu
Nepal<EMAIL_ADDRESS>/<EMAIL_ADDRESS>and  Ajaya Singh
Central Department of Mathematics, Institute of Science and Technology,
Tribhuvan University, Kirtipur, Kathmandu, Nepal<EMAIL_ADDRESS>/
<EMAIL_ADDRESS>
###### Abstract.

In holomorphic semigroup dynamics, Julia set is in general backward invariant
and so some fundamental results of classical complex dynamics can not be
generalized to semigroup dynamics. In this paper, we define completely
invariant Julia set of transcendental semigroup and we see how far the results
of classical transcendental dynamics generalized to transcendental semigroup
dynamics.

###### Key words and phrases:

Rational semigroup, transcendental semigroup, Julia set, completely invariant
Julia set, completely invariant Fatou set.

###### 2010 Mathematics Subject Classification:

37F10, 30D05

This research work of first author is supported by PhD faculty fellowship from
University Grants Commission, Nepal.

##  1\. Introduction

Let us denote the class of all rational maps on $\mathbb{C_{\infty}}$ by
$\mathscr{R}$ and class of all transcendental entire maps on $\mathbb{C}$ by
$\mathscr{E}$. Our particular interest is to study of dynamics of the families
of above two classes of complex analytic (holomorphic) maps. For a collection
$\mathscr{F}=\\{f_{\alpha}\\}_{\alpha\in\Delta}$ of such maps, let

$S=\langle f_{\alpha}\rangle$

be a holomorphic semigroup generated by them. Here $\mathscr{F}$ is either a
collection $\mathscr{R}$ of rational maps or a collection $\mathscr{E}$ of
transcendental entire maps. $\Delta$ is an index set to which $\alpha$ belongs
is finite or infinite. Each $f\in S$ is constructed through the composition of
finite number of functions $f_{\alpha_{k}},\;(k=1,2,3,\ldots,m)$. That is,
$f=f_{\alpha_{1}}\circ f_{\alpha_{2}}\circ f_{\alpha_{3}}\circ\cdots\circ
f_{\alpha_{m}}$. In particular, if $f_{\alpha}\in\mathscr{R}$, we say
$S=\langle f_{\alpha}\rangle$ a rational semigroup and if
$f_{\alpha}\in\mathscr{E}$, we say $S=\langle f_{\alpha}\rangle$ a
transcendental semigroup.

A semigroup generated by finitely many holomorphic functions
$f_{i},(i=1,2,\ldots,\\\ n)$ is called finitely generated holomorphic
semigroup. We write $S=\langle f_{1},f_{2},\ldots,f_{n}\rangle$. If $S$ is
generated by only one holomorphic function $f$, then $S$ is cyclic semigroup.
We write $S=\langle f\rangle$. In this case, each $g\in S$ can be written as
$g=f^{n}$, where $f^{n}$ is the nth iterates of $f$ with itself. Note that in
our study of semigroup dynamics, we say $S=\langle f\rangle$ a trivial
semigroup.

In classical complex dynamics, each of Fatou set and Julia set are defined in
two different but equivalent ways. In first definition, Fatou set is defined
as the set of normality of the iterates of given function and Julia set is
defined as the complement of the Fatou set. The second definition of Fatou set
is given as a largest completely invariant open set and Julia set is given as
a smallest completely invariant close set. Each of these definitions can be
naturally extended to the settings of holomorphic semigroup $S$. In this
paper, we see that these extension definitions are not equivalent in
holomorphic semigroup dynamics. Based on above first definition (that is, on
the Fatou-Julia theory of a complex analytic function), the Fatou set and
Julia set are defined as follows.

######  Definition 1.1 (Fatou set, Julia set).

Fatou set of the semigroup $S$ is defined by

$F(S)=\\{z\in\mathbb{C}:S\;\ \textrm{is normal in a neighborhood of}\;\ z\\}$

and the Julia set $J(S)$ of $S$ is its compliment. Any maximally connected
subset $U$ of the Fatou set $F(S)$ is called Fatou component.

If $S=\langle f\rangle$, then $F(S),J(S)$ and $I(S)$ are respectively the
Fatou set and Julia set in classical complex dynamics. In this situation we
simply write: $F(f)$ and $J(f)$. From the definition 1.1, it is clear that
$F(S)$ is the open set and therefore, it complement $J(f)$ is closed set.
Indeed, these definitions generalize the definitions of Julia set and Fatou
set of the iteration of single holomorphic map.

The fundamental contrast between classical complex dynamics and semigroup
dynamics appears by different algebraic structure of corresponding semigroups.
In fact, non-trivial semigroup (rational or transcendental) need not be, and
most often will not be abelian. However, trivial semigroup is cyclic and
therefore abelian. As we discussed before, classical complex dynamics is a
dynamical study of trivial (cyclic) semigroup whereas semigroup dynamics is a
dynamical study of non-trivial semigroup.

Note that for any holomorphic semigroup $S$, we have

  1. (1)

$F(S)\subset F(f)$ for all $f\in S$ and hence $F(S)\subset\bigcap_{f\in
S}F(f)$.

  2. (2)

$J(f)\subset J(S)$ for all $f\in S$.

######  Definition 1.2 (Forward, backward and completely invariant set).

Let $f$ is a map of a set $X$ into itself. A subset $U\subset X$ is said to be

  1. (1)

forward invariant under $f$ if $f(U)\subset U$;

  2. (2)

backward invariant under $f$ if $f^{-1}(U)=\\{z\in\mathbb{C}:f(z)\in
U\\}\subset U$;

  3. (3)

completely invariant under $f$ if it is both forward and backward invariant.

It is well known that Fatou set $F(f)$ and Julia Set $J(f)$ of any holomorphic
map $f$ are completely invariant. If $S$ is a rational semigroup, then
Hinkkanen and Martin [1, Theorem 2.1] proved the following result.

######  Theorem 1.1.

The Fatou set $F(S)$ is forward invariant under each element of $S$ and Julia
set $J(S)$ is backward invariant for each element of $S$.

If $S$ is a transcendental semigroup, then K. K. Poon [3, Theorem 2.1] proved
the following result.

######  Theorem 1.2.

The Fatou set $F(S)$ is forward invariant for each element of $S$ and Julia
set $J(S)$ is backward invariant for each element of $S$.

Indeed, theorems 1.1 and 1.2 are contrasting to classical complex dynamics. We
generalize the classical completely invariant notion of Fatou set and Julia
set of single function to the completely invariant notion of these sets in
semigroup dynamics. That is, we make completely invariant Fatou set and Julia
set of semigroup $S$ in different way under each element of $S$. In rational
semigroup dynamics, Rich Stankewitz [4] and Rich Stankewitz, Toshiyori Sugawa
and Hiroki Sumi [6] studied completely invariant Julia set and Fatou set. They
([4, Definition 3] and [6, Definitions 1.3 and 1.4]) defined it as follows.

Let $S$ be a rational semigroup. The completely invariant Julia set of $S$ is
the set of the form

$E(S)=\bigcap\\{G:G\;\text{is closed, completely invariant under each}\;g\in
S,\\#(G)\geqslant 3\\}$

where $\\#(G)$ denote the cardinality of $G$. The completely invariant Fatou
set $W(S)$ of $S$ to be the complement of $E(S)$. That is,
$W(S)=\mathbb{C_{\infty}}-E(S)$.

By the similar fashion, we can think completely invariant Julia set and Fatou
set of transcendental semigoup.

######  Definition 1.3.

Let $S$ be a transcendental semigroup. We define completely invariant Julia
set by

$E(S)=\bigcap\\{E:E\;\text{is closed, completely invariant set under
each}\;f\in S\\}$

The completely invariant Fatou set $W(S)$ is defined as the complement of
$E(S)$ in $\mathbb{C}$. That is, $W(S)=\mathbb{C}-E(S)$.

Note that in transcendental semigroup $S$, $E(S)$ exists, is closed and
completely invariant under each $f\in S$ and it contains the Julia set of each
element of $S$. The corresponding Fatou set is open, completely invariant and
contained in the Fatou set of each element of $S$. We prove the following
result which is a generalization of classical complex dynamics to semigroup
dynamics.

######  Theorem 1.3.

If $E(S)$ has non-empty interior, then $E(S)=\mathbb{C}$.

Indeed, this theorem 1.3 is a good connection between classical transcendental
dynamics and transcendental semigroup dynamics.

The set $E(S)$ just we defined above in the definition 1.3 may and may not be
the set $J(S)$. The following examples can help to compere the sets $E(S)$ and
$J(S)$.

######  Example 1.1.

Let $S=\langle f,g\rangle$ be a transcendental semigroup generated by
$f(z)=\lambda\sin z$ and $g(z)=\lambda\sin z+2\pi$. Then $E(S)=J(f)=J(g)$. It
is also verified that $J(S)=J(f)=J(g)$. In this case, $E(S)=J(S)$ and so
$W(S)=F(S)$

######  Example 1.2.

Let $S=\langle f,g\rangle$ be a transcendental semigroup generated by
$f(z)=\lambda e^{z},(0<\lambda<e^{-1})$ and $g(z)=z-e^{z}+1+2\pi i$. Then
$J(f)$ is a Cantor bouquet and $J(g)$ is different from Cantor bouquet. Both
of the functions $f$ and $g$ have a completely invariant Fatou component. In
this case,

In this paper, we prove the following result that will strengthen the examples
1.1 and 1.2.

######  Theorem 1.4.

Let $S=\langle f,g\rangle$ be a semigroup generated by two transcendental
entire functions $f$ and $g$ such that $J(f)\neq J(g)$. If each function $f$
and $g$ has a completely invariant Fatou component, then $E(S)=\mathbb{C}$.

##  2\. Results from rational semigroup dynamics

Stankewitz [4] compered the sets $J(S)$ and $E(S)$ from the reference of the
following two examples.

######  Example 2.1.

[4, Example-1] Suppose that $S=\langle f,g\rangle$ and $J(f)=J(g)$. Then
$E(S)=J(f)=J(g)$ as $J(f)$ is completely invariant under $f$ and $J(g)$ is
completely invariant under $g$. Also, if $J(f)=J(g)$, then $J(S)=J(f)=J(g)$.
So as complement of $J(S)$ and $E(S)$, we have $F(S)=W(S)$.

######  Example 2.2.

[4, Example-2] Let $S=\langle z^{2},z^{2}/a\rangle$, where
$a\in\mathbb{C},|a|>1$. Then Julia set
$J(S)=\\{z:1\leqslant|z|\leqslant|a|\\}$ which not forward invariant. So
$E(S)\neq J(S)$. In this case, $E(S)=\mathbb{C_{\infty}}$. Note that
$J(f)=\\{z:|z|=1\\}$ and $J(g)=\\{z:|z|=|a|\\}$. The Fatou set
$F(S)=\\{z:|z|<1\;\text{or}\;|z|>|a|\\}$ is not S-backward invariant and so
$F(S)\neq W(S)$. In this case, it is obvious that $W(S)=\emptyset$.

On the basis of these two examples, Stankewitz [4] concluded that there are
only two possibilities (as mentioned in examples 3.1 and 3.2) for any
polynomial semigroups. As a conclusion, he proved the following two theorems
([4, Theorems 1 and 2]).

######  Theorem 2.1.

Let $S=\langle f,g\rangle$ be a semigroup generated by two polynomials $f$ and
$g$ of degree at least two. If $J(f)\neq J(g)$, then
$E(S)=\mathbb{C_{\infty}}$.

######  Theorem 2.2.

Let $S^{{}^{\prime}}$ be a rational semigroup which contains two polynomials
$f$ and $g$ of degree at least two. If $J(f)\neq J(g)$, then
$E(S^{{}^{\prime}})=\mathbb{C_{\infty}}$

One of the main result in classical complex dynamics is that if Julia set has
non-empty interior, then Julia set explodes and it becomes whole complex
plane. This result is generalized to completely invariant Julia set $E(S)$.
That is, Stankewitz [4] proved the following result which shows a connection
between classical complex dynamics and semigroup dynamics ([4, Lemma 2]).

######  Theorem 2.3.

If $E(S)$ has non-empty interior, then $E(S)=\mathbb{C_{\infty}}$.

##  3\. Comparison of sets $E(S)$ and $J(S)$ and proof of the main results

In rational semigroup and in particular in polynomial semigroups, there are
lot of studies over such completely invariant Fatou sets and Julia sets (see
for instance in [4, 5, 6] for more detail) but there is nothing study over
transcendental semigroup. So, in this paper, we concentrate more on completely
invariant Julia set of transcendental semigroup $S$.

Next, we workout some constructions for the comparison of sets $E(S)$ and
$J(S)$ if semigroup $S$ is generated by two transcendental entire functions.
Let $S=\langle f,g\rangle$ be a semigroup generated by transcendental entire
functions $f$ and $g$. Note that $J(h)\subset E(S)$ for all $h\in S$ and so
$\bigcup_{h\in S}J(h)\subset E(S)$. Let us define the following countable
collections of sets:

$\mathscr{E}_{0}=\\{J(h)\\}\;\text{for all}\;h\in S$
$\mathscr{E}_{1}=\bigcup_{h\in S}h^{-1}(\mathscr{E}_{0})\cup\bigcup_{h\in
S}h(\mathscr{E}_{0})$

$\ldots\;\;\;\;\;\;\ldots\;\;\;\;\;\;\ldots\;\;\;\;\;\;\ldots\;\;\;\;\;\;\ldots$

$\mathscr{E}_{n+1}=\bigcup_{h\in S}h^{-1}(\mathscr{E}_{n})\cup\bigcup_{h\in
S}h(\mathscr{E}_{n})$

and

$\mathscr{E}=\bigcup_{n=0}^{\infty}\mathscr{E}_{n}$

where $h^{-1}(\mathscr{A})=\\{h^{-1}(A):A\in\mathscr{A}\\}$ and
$h(\mathscr{A})=\\{h(A):A\in\mathscr{A}\\}$ for any collection of sets
$\mathscr{A}$ and a function $h$. The following result will be the convenient
description of the set $E(S)$ of a transcendental semigroup $S$.

######  Theorem 3.1.

For a transcendental semigroup $S=\langle f,g\rangle$, we have
$E(S)=\overline{\bigcup_{A\in\mathscr{E}}A}$.

###### Proof.

By the definition of $E(S)$, it is closed, is completely invariant under each
$h\in S$ and contains $J(h)$ for all $h\in S$. So we can write

$E(S)\supset\overline{\bigcup_{A\in\mathscr{E}}A}$

Since the set $\overline{\bigcup_{A\in\mathscr{E}}A}$ is closed and contains
$J(h)$ for all $h\in S$, it remains to show that it is also completely
invariant under each $h\in S$. Since $h$ is a continuous closed map, so under
each $h\in S$, $h(\overline{\bigcup_{A\in\mathscr{E}}A})$ and
$h^{-1}(\overline{\bigcup_{A\in\mathscr{E}}A})$ are closed sets. It proves our
claim. ∎

######  Corollary 3.1.

The set $E(S)$ is perfect.

###### Proof.

Since $J(h)\subset E(S)$ for all $h\in S$ and $J(h)$ is perfect, unbounded and
contains an infinite number of points for each $h\in S$. This corollary will
be proved if we show $E(S)$ has no isolated points. Suppose $\alpha\in E(S)$
is an isolated point. Then it an isolated point of some $A\in\mathscr{E}$.
Choose a neighborhood $U$ of $\alpha$ so that $U-\\{\alpha\\}\subset W(S)$
where $W(S)$ is completely invariant Fatou set of $S$. Since
$h^{-1}(W(S))\subset W(S)$ and $h(W(S))\subset W(S)$ for all $h\in S$, so each
$h\in S$ omits $E(S)$ on $U-\\{\alpha\\}$, which implies that every element in
$S$ is normal on $U$. Which is a contradiction. ∎

###### Proof of the theorem 1.3.

Let $E(S)^{\circ}\neq\emptyset$, where $E(S)^{\circ}$ denotes the interior of
$E(S)$. Then there exists a disk $D=\\{|z-z_{0}|<r\\}\subset E(S)$ such that
it intersects $J(h)$ for some $h\in S$. Then by [2, Theorem 3.9], for each
finite value $a$, there is sequence $z_{k}\to z_{0}\in J(h)$ and a sequence of
positive integers $n_{k}\to\infty$ such that
$f^{n_{k}}(z_{k})=a,(k=1,2,3,\ldots)$ except at most for a finite value. Then
by backward invariance of $J(h)$, $z_{k}\in J(h)$ and by forward invariance of
$J(h)$, $a\in J(h)$. It shows that every finite value is in $J(h)$, except at
most a single value. Since $h\in S$ is arbitrary, so we must have
$E(S)=\mathbb{C}$. ∎

######  Corollary 3.2.

If $E(S)\neq\mathbb{C}$, then $W(S)$ is unbounded.

###### Proof.

If $W(S)$ is bounded, then $E(S)$ has interior points. By the above theorem,
$E(S)=\mathbb{C}$, which is a contradiction. ∎

Similar to the description of the sets $E(S)$ in lemma 3.1, we can give
analogous description of the Julia set $J(S)$ of transcendental semigroup $S$.
Let us define the following countable collections of sets:

$\mathscr{F}_{0}=\\{J(h)\\}\;\text{for all}\;h\in S$
$\mathscr{F}_{1}=\bigcup_{h\in S}h^{-1}(\mathscr{F}_{0})$

$\ldots\;\;\;\;\ldots\;\;\;\;\;\ldots$

$\mathscr{E}_{n+1}=\bigcup_{h\in S}h^{-1}(\mathscr{F}_{n}))$

and

$\mathscr{F}=\bigcup_{n=0}^{\infty}\mathscr{F}_{n}$

where $h^{-1}(\mathscr{A})=\\{h^{-1}(A):A\in\mathscr{A}\\}$ for any collection
of sets $\mathscr{A}$ and a function $h$. The following result will be the
convenient description of the set $J(S)$ of transcendental semigroup $S$.

######  Theorem 3.2.

For a transcendental semigroup $S=\langle f,g\rangle$, we have
$J(S)=\overline{\bigcup_{A\in\mathscr{F}}A}$.

###### Proof.

By the definition of $J(S)$, it is closed, is backward invariant under each
$h\in S$ and contains $J(h)$ for all $h\in S$. So we can write

$J(S)\supset\overline{\bigcup_{A\in\mathscr{F}}A}$

Since the set $\overline{\bigcup_{A\in\mathscr{F}}A}$ is closed and contains
$J(h)$ for all $h\in S$, it remains to show that it is also backward invariant
under each $h\in S$. Since $h$ is a continuous closed map, so under each $h\in
S$, $h^{-1}(\overline{\bigcup_{A\in\mathscr{F}}A})$ is a closed set. It proves
our claim. ∎

######  Corollary 3.3.

For a transcendental semigroup $S=\langle f,g\rangle$, we have $J(S)\subset
E(S)$.

###### Proof.

Since by construction $\mathscr{F}\subset\mathscr{E}$, so by theorems 3.1,
3.2, assertion of the corollary follows. ∎

######  Corollary 3.4.

If $J(S)$ has non-empty interior, then $E(S)=\mathbb{C}$.

###### Proof.

This corollary follows from theorem 1.3 and corollary 3.3. ∎

Finally we prove our main result that we proposed in the theorem 1.4.

###### Proof of the Theorem 1.4.

Let $U$ be a completely invariant component of $F(f)$. Then by [2, Theorem
4.36], $U$ is unbounded, simply connected and $\partial U=J(f)$. Likewise, a
completely invariant component $V$ of $F(g)$ is unbounded, simply connected
and $\partial V=J(g)$. $J(f)\neq J(g)$ implies that $\partial U\neq\partial
V$. By [2, Theorem 3.8], $J(f)$ and $J(g)$ are unbounded, so $U\cap
J(g)\neq\emptyset$ and $V\cap J(f)\neq\emptyset$. The fact $\partial
U\neq\partial V$ implies that $J(f)$ must intersect interior of $V$ and $J(g)$
must intersect interior of $U$.

Let $z\in J(f)\cap V^{\circ}$, where $V^{\circ}$ is an interior of $V$. Then
by the forward invariance of $E(S)$ and $V^{\circ}$ under the map $g$, so, its
nth iterates, that is, $g^{n}(z)\in E(S)$ and $g^{n}(z)\in V^{\circ}$ for all
$n\in\mathbb{N}$. Likewise, we can write $f^{n}(z)\in E(S)$ and $f^{n}(z)\in
U^{\circ}$ for all $n\in\mathbb{N}$. This shows that $E(S)$ intersects open
sets $U^{\circ}$ and $V^{\circ}$. So, $E(S)$ intersects $U^{\circ}\cap
V^{\circ}$. Since $E(S)$ is perfect and completely invariant set, so it
contains all limits of the sequences $(f^{n})$ and $(g^{n})$. This prove that
$E^{\circ}\neq\emptyset$ and hence $E(S)=\mathbb{C}$. ∎

## References

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