# Nature of long-lived moiré interlayer excitons in electrically tunable
MoS2/MoSe2 heterobilayers

Evgeny M. Alexeev  [<EMAIL_ADDRESS>Carola M. Purser  [  Carmem M. Gilardoni
[  James Kerfoot  Hao Chen  [  Alisson R. Cadore  [  Bárbara L.T. Rosa  [
Matthew S. G. Feuer  Evans Javary  [  Patrick Hays  [  Kenji Watanabe  [
Takashi Taniguchi  [  Seth Ariel Tongay  [  Dhiren M. Kara  [  Mete Atatüre  [
<EMAIL_ADDRESS>Andrea C. Ferrari  [<EMAIL_ADDRESS>
###### Abstract

Interlayer excitons in transition-metal dichalcogenide heterobilayers combine
high binding energy and valley-contrasting physics with long optical lifetime
and strong dipolar character. Their permanent electric dipole enables
electric-field control of emission energy, lifetime, and location. Device
material and geometry impacts the nature of the interlayer excitons via their
real- and momentum-space configurations. Here, we show that interlayer
excitons in MoS2/MoSe2 heterobilayers are formed by charge carriers residing
at the Brillouin zone edges, with negligible interlayer hybridization. We find
that the moiré superlattice leads to the reversal of the valley-dependent
optical selection rules, yielding a positively valued g-factor and cross-
polarized photoluminescence. Time-resolved photoluminescence measurements
reveal that the interlayer exciton population retains the optically induced
valley polarization throughout its microsecond-long lifetime. The combination
of long optical lifetime and valley polarization retention makes MoS2/MoSe2
heterobilayers a promising platform for studying fundamental bosonic
interactions and developing excitonic circuits for optical information
processing.

###### keywords:

van der Waals heterostructures, transition metal dichalcogenides, interlayer
excitons, moiré superlattice, valley polarization, Stark shift,
photoluminescence

University of Cambridge]Cambridge Graphene Centre, University of Cambridge, 9
JJ Thomson Avenue, CB3 0FA, Cambridge, UK [University of Cambridge]Cavendish
Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK
University of Cambridge]Cavendish Laboratory, University of Cambridge, JJ
Thomson Avenue, Cambridge CB3 0HE, UK [University of Cambridge]Cambridge
Graphene Centre, University of Cambridge, 9 JJ Thomson Avenue, CB3 0FA,
Cambridge, UK University of Cambridge]Cavendish Laboratory, University of
Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK University of
Cambridge]Cambridge Graphene Centre, University of Cambridge, 9 JJ Thomson
Avenue, CB3 0FA, Cambridge, UK University of Cambridge]Cambridge Graphene
Centre, University of Cambridge, 9 JJ Thomson Avenue, CB3 0FA, Cambridge, UK
[Brazilian Nanotechnology National Laboratory]Brazilian Nanotechnology
National Laboratory, Brazilian Center for Research in Energy and Materials,
Sao Paulo, Brazil University of Cambridge]Cambridge Graphene Centre,
University of Cambridge, 9 JJ Thomson Avenue, CB3 0FA, Cambridge, UK
University of Cambridge]Cavendish Laboratory, University of Cambridge, JJ
Thomson Avenue, Cambridge CB3 0HE, UK [ENS PAris]École Normale Supérieure,
PSL, 5 Rue D’ulm Paris, 75005, France Arizona State University]Materials
Science and Engineering, School for Engineering of Matter,Transport and
Energy, Arizona State University, Tempe, Arizona 85287, United States National
Institute for Materials Science]Research Center for Electronic and Optical
Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba
305-0044, Japan National Institute for Materials Science]Research Center for
Materials Nanoarchitectonics, National Institute for Materials Science, 1-1
Namiki, Tsukuba 305-0044, Japan Arizona State University]Materials Science and
Engineering, School for Engineering of Matter,Transport and Energy, Arizona
State University, Tempe, Arizona 85287, United States University of
Cambridge]Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue,
Cambridge CB3 0HE, UK University of Cambridge]Cavendish Laboratory, University
of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK University of
Cambridge]Cambridge Graphene Centre, University of Cambridge, 9 JJ Thomson
Avenue, CB3 0FA, Cambridge, UK

Van der Waals heterostructures comprising monolayer transition-metal
dichalcogenides (TMDs) have emerged as a promising platform for
optoelectronics1, 2, 3, 4 and quantum technology5 as they combine optically
addressable spin and valley degrees of freedom6, 7, 8 with unique tunability
through the choice of material combination9, 10, 11 and rotational
alignment12, 13, 14. TMD heterobilayers have drawn particular interest due to
their ability to host interlayer excitons (iXs)15, 16 which offer lifetime
approaching 200 $\mu$s17, strong repulsive dipolar interaction18, 19, and high
sensitivity to rotational alignment20, 21, 22, strain23, 17, electric24 and
magnetic25 fields. Different TMD combinations give rise to iX with drastically
different properties, including oscillator strength26, 27, center-of-mass
momentum21, 20, and degree of interlayer hybridization19, 22. Of the plethora
of possible TMD combinations, the majority of research effort focused on
MoSe2/WSe228, 29, 30, 27 and WS2/WSe231, 32, 33 heterobilayers. For other
material combinations, key aspects of iX nature, such as real- and momentum-
space configuration, remain elusive due to the complexity of the underlying
physics.

In this work, we investigate iX in MoS2/MoSe2 heterobilayers using
polarization-resolved magneto-photoluminescence spectroscopy. We find that iX
photoluminescence (PL) is visible only in devices with relative twist angle
less than 5∘. This indicates that the constituent iX charge carriers reside at
the edges of the Brillouin zone. We study the iX PL response to out-of-plane
electric and magnetic fields and show that iX is formed by charge carriers at
the $\pm$K valleys with negligible degree of interlayer hybridization. Our
time- and polarization-resolved PL measurements reveal microsecond-scale
retention of optically induced valley polarization, demonstrating the
potential of MoS2/MoSe2 heterobilayers for opto-valleytronic applications.

Figure 1: iX in MoS2/MoSe2 heterobilayers. (a) Optical microscope image of an
electrically tunable MoS2/MoSe2 heterobilayer device. Monolayer MoS2 and MoSe2
regions are outlined in red and green, respectively. Solid (dashed) black line
shows the position of top (bottom) gate composed of few-layer graphene (FLG)
(b) Cross-sectional schematic of the device structure, indicating spatial
configuration of iX. (c) Room-temperature PL spectra recorded in two devices
with different rotational alignment. The closely rotationally aligned device
(top panel, $\theta=1^{\circ}$) shows intralayer MoS2 B and A excitons and
MoSe2 A exciton peaks, as well as an iX peak appearing in a lower-energy range
(highlighted in copper) that is not visible in the PL spectrum of the strongly
misaligned device (bottom panel, $\theta=28^{\circ}$). (d) Polarization-
resolved SHG intensity recorded in the isolated monolayer (red) MoS2 and
(green) MoSe2 regions of the two devices, confirming the twist angle of
$\theta=1^{\circ}$ (top) and $\theta=28^{\circ}$ (bottom).

Figure 1a shows an optical microscope image of one of our electric-field
tunable MoS2/MoSe2 heterobilayers, with panel b displaying its cross-section.
The hexagonal boron nitride layers provide a flat and clean dielectric
environment for the MoS2/MoSe2 heterobilayer, and the transparent few-layer
graphene top and bottom gates allow optical measurements under applied out-of-
plane electric field. Each of the eight devices is fabricated from
micromechanically exfoliated crystals using deterministic dry mechanical
transfer34, 35, and thickness and quality of constituent layers are
characterized using Raman spectroscopy36 (see Methods and SI Fig. S1). The
devices offer a range of twist angles between the TMD monolayers $\theta$,
enabling the investigation of iX momentum-space configuration. Figure 1c
compares room-temperature PL spectra of two devices with twist angle
$\theta=1^{\circ}$ (top) and $\theta=28^{\circ}$ (bottom). We identify
$\theta$ using polarization-resolved second-harmonic generation (SHG)
measurements (Fig. 1d). Both devices show PL peaks corresponding to the A
exciton in MoSe2 (MoSe2 XA) at 1.55 eV and the A and B excitons in MoS2 (MoS2
XA and XB) at 1.85 and 2.0 eV, respectively37, 38. Crucially, the iX PL peak
at $\sim$1.3 eV is visible only in the device with $\theta=1^{\circ}$. Of the
eight devices with twist angles ranging from 1∘ to 28∘ that we studied, only
those with $\theta\leq 5^{\circ}$ reveal the iX PL peak at room temperature
(SI Fig. S2), in line with a recent report39. Thus, close rotational alignment
is critical for the observation of iX in MoS2/MoSe2 heterobilayers.

The $\sim$3.7% mismatch in lattice constants of MoS2 and MoSe240 eliminates
twist-angle dependence of the interlayer distance as an underlying source of
this behavior41. Instead, the high sensitivity of the iX PL intensity to
$\theta$ indicates that the iX is formed by the charge carriers residing in
valleys at the edges of the Brillouin zone. Homo- and heterobilayers where at
least one of the charge carriers resides at the $\Gamma$ valley at the center
of the Brillouin zone display iX PL throughout the entire $\theta$ range, as
the momentum-space separation between electron and hole remains unchanged42,
21, 43. In contrast, in heterobilayers where both constituent charges reside
at the edges of the Brillouin zone, momentum-space separation of electron and
hole suppresses radiative recombination of iX in devices with $\theta$ away
from 0 or $60^{\circ}$20, 13, consistent with our observations.

Figure 2: Electric field tuning of iX. (a) Normalized iX PL under applied out-
of-plane electric field. iX PL energy shows a linear shift with the slope of
$\sim$0.31 eV nm V$-1$, corresponding to the dipole size of $d=0.56(3)$ nm.
The dipole size closely matches the expected interlayer distance. (b)
Variation of iX PL decay time as a function of applied electric field. Gray
and black circles correspond to the fast ($\tau_{1}$) and slow ($\tau_{2}$)
time constants, respectively. (c) PL decay acquired at +0.24, 0, and -0.24 V
nm-1 applied electric field. Red curves are bi-exponential fits to the data.

We next identify the real-space configuration of iX by studying its response
to applied out-of-plane electric field. Figure 2a presents the normalized iX
PL spectrum recorded as a function of electric field at 4-K temperature. The
iX PL energy shifts linearly with a rate of $\sim$0.31 eV nm V$-1$ and can be
tuned over a 144-meV range within the gate tuning limits of our device. We
find an average tuning response across three devices of $\sim$0.30 eV nm
V$-1$, yielding an average dipole size of 0.55(3) nm 19, 24 (SI Fig. S3). This
value is in good agreement with the $\sim$0.6-nm separation between the
layers44. A similar dipole size was observed in MoSe2/WSe219 heterobilayers
with iX is formed by non-hybridized electrons and holes, while MoSe2
homobilayers show a reduced dipole size of 0.26 nm due to charge-carrier
hybridization45. Comparatively, our results suggest negligible interlayer
hybridization for our devices.

Figure 2b shows the iX PL decay time constants as a function of the applied
electric field. We extract these constants from a bi-exponential fit to the
time-resolved PL. Figure 2c presents examples of PL decay trace recorded at
three applied field values along with their corresponding fit curves. We
observe microsecond-long iX lifetime, with a fast time constant
$\tau_{1}=1.0(1)~{}\mu$s and a slow time constant $\tau_{2}=4.4(4)~{}\mu$s at
zero electric field – an order of magnitude longer than typical lifetimes of
10-100 ns reported for MoSe2/WSe2 heterobilayers13. The slow time constant in
other devices ranges from 0.1 $\mu$s to 3.0 $\mu$s (SI Fig. S4). The fast time
constant is mostly field-independent, except for $\sim$0.06 V nm$-1$, where it
increases to 1.3 $\mu$s. Shortening of $\tau_{1}$ for electric field away from
this value is likely caused by inadvertent electrostatic doping induced by a
slight asymmetry in the thicknesses of the bottom and top dielectric layers.
The slow time constant $\tau_{2}$ shows a gradual decrease with increasing
electric field, consistent with a change in radiative lifetime due to a field-
induced variation of electron-hole separation. For electric field anti-
parallel to the electric dipole moment of the iX, the separation between the
two charge carriers is reduced, leading to an increased probability of their
radiative recombination. The opposite process takes place for the parallel
field alignment. That said, the PL decay time remains slow ($\tau_{1}\geq
0.04~{}\mu$s, $\tau_{2}\geq 1.9~{}\mu$s) throughout the entire field-tuning
range.

Figure 3: Magneto-PL spectroscopy of iX. (a) Helicity-resolved iX PL spectra
recorded under applied out-of-plane magnetic field ranging from -6 to 6 T
using $\sigma^{+}$ polarized 1.94 eV optical excitation; blue (red) curves
correspond to PL with $\sigma^{+}$ ($\sigma^{-}$) polarization. (b) Energy
splitting between $\sigma^{+}$ and $\sigma^{-}$ polarized PL as a function of
out-of-plane magnetic field for (top) iX in heterobilayer and (bottom) neutral
excitons (X0) in MoSe2 monolayer regions. Landé g-factors extracted using
linear fits are listed next to each plot. (c) Optically induced valley
polarization calculated as
$\rho_{\mathrm{opt}}=(I^{++}+I^{--}-I^{+-}-I^{-+})/(I^{++}+I^{--}+I^{+-}+I^{-+})$
as a function of magnetic field for iX (top) and MoSe2 X0 (bottom), where
$I^{XY}$ is the intensity of PL with $\sigma^{Y}$-polarization collected under
$\sigma^{X}$-polarized excitation. Unfilled circles in the panel and the inset
are extracted from a fine scan around 0 T, showing the small-field dependence
of $\rho_{\mathrm{opt}}$ for iX.

We use polarization-resolved magneto-PL spectroscopy to identify valley
configuration of iX. Figure 3a shows iX PL spectra recorded using right
circularly polarized ($\sigma^{+}$), 1.94-eV excitation as a function of
applied out-of-plane magnetic field $B$ ranging from -6 to 6 T; blue (red)
curves correspond to PL with $\sigma^{+}$ ($\sigma^{-}$) polarization. The iX
PL remains cross-polarized with respect to the excitation laser throughout the
entire magnetic field range. Two distinct mechanisms are known to give rise to
this behavior in TMD heterobilayers: the directional intervalley scattering 46
and the moiré-induced reversal of the valley-dependent optical selection rules
47.

We identify the underlying mechanism in MoS2/MoSe2 heterobilayers based on the
sign of the energy splitting between $\sigma^{+}$ and $\sigma^{-}$ polarized
PL under applied magnetic field. Figure 3b is a plot of the energy splitting
($\Delta E$) as a function of $B$ for the iX in the heterobilayer region (top
panel) and the neutral intralayer excitons (X0) in an isolated monolayer MoSe2
region (bottom panel). We define $\Delta E$ as
$E_{\sigma^{+}}-E_{\sigma^{-}}=g\mu_{\mathrm{B}}B$, where
$E_{\sigma^{+}}$($E_{\sigma^{-}}$) is the energy of the $\sigma^{+}$
($\sigma^{-}$) polarized PL, $g$ is the effective Landé g-factor and
$\mu_{\mathrm{B}}$ is the Bohr magneton. For X0 in MoSe2, we extract
$g=-3.90(2)$, consistent with previous reports48, where the minus sign stems
from valley-Zeeman interaction and valley-dependent optical selection rules
for TMD monolayers49: $\sigma^{+}$-polarized light couples to optical
transitions in the +K valley, which has lower energy at positive $B$. In
contrast, we obtain $g=+2.50(7)$ for iX - the positive sign of $g$ shows that
iX PL from the +K valley appears with $\sigma^{-}$ polarization, confirming
the reversal of optical selection rules with respect to the monolayer case.
All devices showed positive iX g-factors, with an average value of +4.5. In
TMD heterobilayers, the reversal of the selection rules arises from local
changes in crystal symmetry induced by the moiré superlattice 47. We observe
the iX g-factors ranging from +1.0 to +8.0, likely due to the difference in
the moiré pattern parameters arising from different twist angles and local
strain in different devices (see SI Fig. S5).

Two mechanisms can give rise to the observed PL polarization under finite
magnetic field, namely the optically induced valley polarization50 and the
Zeeman-shift-induced valley thermalization37. The former is limited by non-
directional intervalley scattering, while the latter arises from exciton
relaxation into the lower-energy valley. We calculate the degree of optically
induced valley polarization independently as
$\rho_{\mathrm{opt}}=\frac{I^{++}+I^{--}-I^{+-}-I^{-+}}{I^{++}+I^{--}+I^{+-}+I^{-+}}$,
where $I^{XY}$ represents the intensity of PL with $\sigma^{Y}$-polarization
collected under $\sigma^{X}$–polarized excitation. Figure 3c displays the
dependence of $\rho_{\mathrm{opt}}$ on the applied magnetic field for iX and
MoSe2 X0. MoSe2 X0 shows a constant PL polarization degree of $\sim$4%,
consistent with earlier reports51, 52. In contrast, $|\rho_{\mathrm{opt}}|$
for iX shows a distinct increase with increasing $|B|$, saturating at $\sim$6
% above $\pm$20 mT (see inset in Fig. 3c). This dependence is consistent
across all devices, with $|\rho_{\mathrm{opt}}|$ ranging from 6% to 14%
maximum value. In the absence of magnetic field, $|\rho_{\mathrm{opt}}|$
ranges from 0% to 7%. Similar sharp changes in valley polarization degree with
applied magnetic field have been observed for iX in MoSe2/WSe253 and
MoS2/WSe254 heterobilayers, as well as intralayer excitons in WS2 and WSe2
monolayers55. This effect was attributed to the suppression of intervalley
scattering of intralayer excitons within the monolayer with dark excitonic
ground state6. However, we observe device-specific saturation field for
$\rho_{\mathrm{opt}}$ ($B_{\mathrm{sat}}$) ranging from 0.02 to 3 T (SI Fig.
S6), indicating that it is not defined by the properties of individual
monolayers, but the collective property of the assembled heterostructure.

Figure 4: Temporal evolution of iX valley polarization. (a) Polarization-
resolved iX PL decay acquired at 0 and 40 mT using $\sigma^{+}$ polarized
excitation; blue (red) curve corresponds to PL intensity co- (cross-)
polarized with the excitation laser; the data sets are offset for clarity and
normalized to the intensity of co-polarized component at zero delay. (b) Time-
resolved changes of $\rho_{\mathrm{opt}}$ for B = 0 mT (black) and B = 40 mT
(gray). The solid curves are guides to the eye.

Figure 4a presents the polarization-resolved decay of iX PL recorded at 0 mT
and 40 mT in a device with $B_{\mathrm{sat}}$$\sim$200 mT. The difference in
intensity for cross-co polarization allows us to monitor
$|\rho_{\mathrm{opt}}|$ as a function of time. Figure 4b plots the time-
resolved $\rho_{\mathrm{opt}}$ extracted from the iX PL decay measured at 0 mT
and 40 mT; the solid curves are guides to the eye. Without magnetic field
(black filled circles), iX has a low polarization degree
($|\rho_{\mathrm{opt}}|<2\%$) throughout the measurement range. In contrast,
at $B=40$ mT (gray filled circles) $|\rho_{\mathrm{opt}}|$ starts at 6% and
shows a gradual decay towards zero, with a characteristic 1/e time of $\sim
2~{}\mu$s. These results indicate that the loss of valley polarization for iX
is governed by at least two processes occurring at different timescales: a
fast intervalley relaxation with a characteristic time shorter than 10 ns
(i.e., timing resolution of our measurement) dominates at zero magnetic field.
This process is suppressed at 40 mT, revealing a slower, microsecond-scale
relaxation. We note that this timescale stems from both the loss of valley
polarization and the decay of exciton population and therefore ultimately it
provides a lower bound to the iX valley lifetime and a measure of the valley-
excitonic information storage time.

In conclusion, we show that the iX in MoS2/MoSe2 heterobilayers is formed by
electrons and holes residing at the edges of the Brillouin zone, with
negligible degree of interlayer hybridization. We find that iX retains its
optically induced valley polarization, with the cross-polarized iX PL stemming
from the moiré-induced reversal of selection rules. Magnetic field enhances
valley-polarization retention by suppressing fast intervalley scattering. The
typical magnetic field required for this ($\leq 200~{}$mT) is within reach of
a variety of readily accessible techniques, including assembling
heterostructures on magnetic substrates56, or using rare-earth magnets57.
Moreover, in some devices we observe $|\rho_{\mathrm{opt}}|$ up to 7% at zero
field, allowing for magnet-free operation. The combination of microsecond-long
iX PL lifetime and the retention of valley polarization offers the prospect of
combining excitonic and valleytronic functionalities in a single
optoelectronic device.

##  0.1 Methods

Sample fabrication. All layers used for the fabrication of the electrically
tunable MoS2/MoSe2 heterobilayer devices were produced by micromechanical
cleavage of bulk crystals. Bulk TMD crystals were prepared by flux zone growth
method58 and bulk hBN crystals were grown by the temperature-gradient
method59. Graphite crystals were sourced from NGS. The thickness of exfoliated
crystals was estimated using optical contrast60 and confirmed using
photoluminescence and Raman spectroscopy for TMDs under 532-nm (2.33-eV) laser
illumination (LabRAM HR Evolution, Horiba) and atomic force microscopy for hBN
(Dimension Icon, Bruker). Electrically tunable heterobilayer devices were
assembled by deterministic dry mechanical transfer using polymer stamps34, 35.
The twist angle between the TMD layers was identified using polarization-
resolved SHG61 measured at room temperature using a custom-built optical
setup. The SHG laser was a Chameleon Compact Optical Parametric Oscillator
providing $\sim$200 fs pulses with a repetition rate of 80 MHz centered at
1320 nm. To minimize chromatic aberrations, a linearly polarized laser beam
with $\sim$5-mW power was focused onto the sample using a 40x reflective
objective (numerical aperture of 0.5, LMM40X-P01, Thorlabs). Polarization
orientation was controlled using a super-achromatic half-wave plate
(SAHWP05M-1700, Thorlabs) mounted in a motorized rotational mount. Electrical
contacts to TMD layers and transparent FLG gates were created by direct laser
lithography (LW-405B+, Microtech) with a positive resist (AZ5214E,
MicroChemicals) followed by electron beam evaporation (PVD200Pro, Kurt J.
Lesker) of 5 nm of Cr followed by 45 nm of Au. The resist excess metal layer
was then lifted off by immersion in acetone and isopropanol for 30 minutes.

Photoluminescence measurements. Room-temperature PL measurements were
performed using LabRAM HR Evolution Raman microscope under 532-nm (2.33-eV)
laser illumination. Helicity-resolved magneto-optical measurements were
performed in a close-cycle bath cryostat (Attodry 1000, Attocube) equipped
with a superconducting magnet at a nominal sample temperature of 4 K.
Excitation and collection light pass through a home-built confocal microscope
in reflection geometry, with a 0.81 numerical aperture apochromatic objective
(LT-307 APO/NIR/0.81, Attocube). The PL measurements were performed using
638-nm (1.94-eV) continuous-wave excitation (MCLS1-638, Thorlabs), with
incident power below 5 $\mu$W. The PL signal collected in epi-direction was
isolated using a long-pass filter (FELH0700, Thorlabs) and detected by a
0.75-m spectrometer (SpectraPro 2750, Princeton Instruments) with 150 l mm-1
grating and a nitrogen cooled CCD camera (Spec-10, Princeton Instruments).
Time-resolved measurements were performed using a single-photon avalanche
photodiode (SPCM-AQRH-16-FC, Excelitas Technologies) and a time-to-digital
converter (quTAU, qutools GmbH) with 81 ps timing resolution. For these
measurements, the intensity of CW laser was modulated using an acousto-optic
modulator (MT350-A0.12-VIS, AA Opto Electronic), producing 200-ns pulses with
the 100-kHz repetition rate. A dual-channel source meter (2612B, Keithley) was
used for electric field tuning.

We acknowledge funding the EU Graphene and Quantum Flagships, ERC Grants
Hetero2D, GSYNCOR, GIPR, EIC Grant CHARM, EPSRC Grants EP/K01711X/1,
EP/K017144/1, EP/N010345/1, and EP/L016087/1. S.A.T acknowledges primary
support from DOE-SC0020653 (materials synthesis). K.W. and T.T. acknowledge
support from the JSPS KAKENHI (Grant Numbers 21H05233 and 23H02052) and World
Premier International Research Center Initiative (WPI), MEXT, Japan. We thank
V. Falko, C. Faugeras, and I. Paradisanos for fruitful discussions.

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# A geometric dynamical system with relation to polygonal billiards

Samuel Everett<EMAIL_ADDRESS>
###### Abstract.

We introduce a geometric dynamical system where iteration is defined as a
cycling composition of a finite collection of geometric maps, which act on a
space composed of three or more lines in $\mathbb{R}^{2}$. This system is
motivated by the dynamics of iterated function systems, as well as billiards
with modified reflection laws. We provide conditions under which this
dynamical system generates periodic orbits, and use this result to prove the
existence of closed nonsmooth curves over $\mathbb{R}^{2}$ which satisfy
particular structural constraints with respect to a space of intersecting
lines in the plane.

###### Key words and phrases:

Piecewise continuous, contraction mapping, periodic orbits, billiards

###### 2010 Mathematics Subject Classification:

37B99, 37E99, 51N20, 37C27

##  1\. Introduction

The theory of mathematical billiards in polygons concerns the uniform motion
of a point mass (billiard) in a polygonal plane domain, with elastic
reflections off the boundary according to the mirror law of reflection: the
angle of incidence equals the angle of reflection. In addition to billiards
obeying the mirror law of reflection, well studied areas include billiards
with modified reflection laws, so that the angle of reflection is some
function of the angle of incidence (see e.g., [arroyo, arroyo2, magno, magno2,
gtroub] and the references therein), and tiling billiards, where trajectories
refract through planar tilings (see [davis, davis2]).

A basic question one can ask is whether there exists a periodic billiard
trajectory. Indeed, a long-standing open question in polygonal billiards with
standard reflection laws is whether every polygon contains a periodic billiard
orbit (see _Problem 10_ in [gutkin], and [gutkint] for a survey); in fact, the
question remains unsolved for particular obtuse triangles. Intense study on
this problem has led to progress (see, e.g. [masur] for results on rational
polygons, and [schwartz, galperin, troub, halb] for results on triangles), and
many deep theorems have been obtained, but the problem remains open.

The aim of this paper is to provide a dynamical system suitable for use in
studying problems related to determining periodic trajectories in polygonal
billiards, and to generalize the system studied in [everett]. The dynamical
system studied here can be analogized to an iterated function system (see
[hutch] for review) where the defining collection of contraction mappings are
geometrically defined over lines in the plane, and composed in a fixed,
cycling order. The system can also be thought of as billiards where
trajectories reflect off or refract through lines as a function of the line
they are incident to.

As an application of this dynamical system, we prove Theorem 1.1, which
asserts existence of nonsmooth closed curves satisfying particular geometric
constraints with respect to a space of lines on the plane. In fact, such
closed curves can coincide with periodic billiard trajectories. We state this
result after giving some notation.

Let $X_{m}\subset\mathbb{R}^{2}$ denote a union of $m\geq 3$ nonconcurrent
lines in $\mathbb{R}^{2}$ where at least one line is not parallel or
perpindicular with any other, and assign each line in $X_{m}$ a unique label
$L_{i}$, $i\in\\{1,2,...,m\\}$. Let $p_{1},p_{2},...,p_{n}$, be a sequence of
$n\geq m$ points in $X_{m}$ such that consecutive points, including $p_{n}$
and $p_{1}$, are distinct, and if $p_{k}\in L_{i}$, then $p_{k+1}\in L_{j}$,
$i\not=j$ (with the convention that $p_{n+1}=p_{1}$). Join consecutive pairs
of such points, including $p_{n}$ and $p_{1}$, with line segments to construct
a _closed curve_ $\Gamma$ over $X_{m}$. Traversal of a closed curve in a fixed
direction allows for construction of an incidence angle sequence
$\theta_{1},\theta_{2},...,\theta_{n}$ with respect to a line sequence
$L_{a_{1}},L_{a_{2}},...,L_{a_{n}}$, by taking the acute or right angle
$\theta_{i}$ between each segment of the closed curve and a line with label
$L_{a_{i}}$ it is incident to, with respect to the traversal direction. Refer
to Figure 1 for visual demonstration.

Figure 1. The values
$\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5},\theta_{6}$ compose an
incidence angle sequence with respect to the line sequence
$L_{3},L_{4},L_{2},L_{4},L_{3},L_{1}$.

######  Theorem 1.1.

For any space $X_{m}$ with labeled lines, let
$\theta_{1},\theta_{2},...,\theta_{n}$, $n\geq m\geq 3$, be any sequence of
acute angles, and let $L_{a_{1}},L_{a_{2}},...,L_{a_{n}}$ be a sequence of
line labels such that no two consecutive labels are the same, including
$L_{a_{n}}$ and $L_{a_{1}}$, and each of the $m$ possible labels occur at
least once in the sequence. Then there exists a closed curve $\Gamma$ over
$X_{m}$ that admits an incidence angle sequence
$\theta_{1},\theta_{2},...,\theta_{n}$ with respect to the line sequence
$L_{a_{1}},L_{a_{2}},...,L_{a_{n}}$ when traversed in a fixed direction.

In the case where a closed curve is strictly contained within a polygon formed
by the intersecting lines composing $X_{m}$, the closed curve does not cross
over any lines in the space, so the angles of incidence implicitly define
angles of reflection. Hence, when the parameters of the closed curve are such
that the angles of incidence equal the angles of reflection, or the angles of
reflection are a function of the angles of incidence, the closed curve
corresponds to a periodic billiard trajectory obeying the mirror law of
reflection or some modified reflection law. However, all closed curves need
not correspond to a periodic billiard trajectory.

This paper is organized into two main parts, separated by study of two related
dynamical systems. In the first, from Sections 2 through 3, we define a
dynamical system that provides controllable and predictable behavior, which we
use to prove Theorem 1.1. In the second part, from Sections 4 through 5, we
redefine components of the dynamical system given in Section 2 in a way that
introduces discontinuities. The introduction of such discontinuities leads to
far more complex dynamics that can share characteristics with piecewise
isometries (see [goetz1, goetz2] for review) and generalizes [everett]. We
prove a theorem that asserts orbits of this system are asymptotically stable
when particular geometric conditions are satisfied.

### Acknowledgements.

The author is grateful to the anonymous referee of this article for their many
helpful suggestions, and to Nikhil Krishnan (Princeton University) for the
valuable feedback and discussions.

##  2\. Preliminaries

Let $L_{i},L_{j}$ label distinct lines in the space $X_{m}$. Then, for every
$x\in L_{i}$ we may determine two lines, $\mathscr{L}_{0},\mathscr{L}_{1}$
such that $\\{x\\}=\mathscr{L}_{0}\cap\mathscr{L}_{1}$ and
$\mathscr{L}_{0},\mathscr{L}_{1}$ intersect with line $L_{j}$ at an angle
$\theta$ in $(0,\pi/2)$, with intersection points $z_{0}$ and $z_{1}$ in
$L_{j}$, respectively. For a visual demonstration, refer to Figure 2. We call
$z_{0},z_{1}$ the orientation 0 and 1, angle $\theta$ projections of $x$ onto
$L_{j}$. If $\theta=\pi/2$, then we call the line intersection point $z$ the
perpendicular projection of $x$ onto $L_{j}$. In the case when $x\in L_{i}\cap
L_{j}$, the projection of $x$ onto $L_{i}$ or $L_{j}$ is simply $x$ itself.

Figure 2. An illustration of orientation 0 and 1, angle $\theta$ projections
of $x$ onto points $z_{0},z_{1}$ in $L_{j}$. Note that an orientation
$o\in\\{0,1\\}$ corresponds with a choice in line $\mathscr{L}_{o}$.

######  Definition 2.1.

Let $\theta\in(0,\pi/2]$, $o\in\\{0,1\\}$, and $i\in\\{1,...,m\\}$. We call a
mapping $r:X_{m}\rightarrow X_{m}$ a _rule_ , if $r(x)$ is an angle $\theta$,
orientation $o$ projection of $x\in X_{m}$ onto a line $L_{i}$ in $X_{m}$. We
may also notate rules as $r(x;\theta,o,L_{i})$ to make the parameters
explicit.

When the rule projection angle $\theta=\pi/2$, we simply write
$r(x;\theta,L_{i})$ when notating rules as there is only one possible
orientation. Figure 3 provides visual demonstration of the composition of two
rules, $r_{1}(x)\coloneqq r_{1}(x;\theta_{1},1,L_{2})$ and $r_{2}(x)\coloneqq
r_{2}(x;\theta_{2},0,L_{3})$ over a point $x_{0}\in L_{1}\subset X_{3}$, so
that

$r_{1}(x_{0})=x_{1}\text{ and }r_{2}(r_{1}(x_{0}))=x_{2}.$

Figure 3. An illustration of the composition of two rules,
$r_{1}(x;\theta_{1},1,L_{2})$ and $r_{2}(x;\theta_{2},0,L_{3})$ over a point
$x_{0}\in L_{1}$, so that $r_{1}(x_{0})=x_{1}$, and $r_{2}(x_{1})=x_{2}$. The
figure also shows the two orientation options for each rule, and the dotted
line corresponds to the $\theta=\pi/2$ case for each rule.

We require rule orientation to be defined in a predictable and consistent way,
so that it is never ambiguous which projection points correspond to which
orientation value. For this paper, we choose a natural and mathematically
convenient convention where the orientation 0 and 1 projection points under a
rule $r(x)$ are the “left” and “right” points, “from the perspective of $x$”.
Figures 3 and 4 demonstrate this convention under point translation.

Figure 4. How orientation is preserved under point translation (from $x$ to
$y$) under a rule with projection angle $\theta$. Note how the projection
lines corresponding with a fixed orientation are antiparallel across line
intersection points, and always map opposite the same line intersection angle.

We define a _rule sequence_ associated to a space $X_{m}$ to be a sequence of
$n\geq m\geq 3$ rules, denoted $\\{r_{i}\\}_{i=1}^{n}$, with the restriction
that consecutive rules in the rule sequence, including $r_{1}$ and $r_{n}$,
cannot map onto the same line in $X_{m}$. Furthermore, we require each line in
$X_{m}$ be mapped onto by at least one of the rules in an associated rule
sequence.

######  Definition 2.2.

Let $T_{n}:X_{m}\rightarrow X_{m}$ denote an _$n$ -rule map_, where iteration
of $T_{n}$ is defined to be a cycling composition of the $n\geq 3$ rules in an
associated _defining rule sequence_ $\\{r_{i}\\}_{i=1}^{n}$. That is, if
$\\{r_{i}\\}_{i=1}^{n}$ is the defining rule sequence for $n$-rule map
$T_{n}$, then for $x\in X_{m}$, we define iteration of $T_{n}$ so that

$T_{n}(T_{n}^{n+1}(x))=T_{n}^{n+2}(x)=r_{2}(r_{1}(r_{n}(...r_{2}(r_{1}(x))))).$

Figure 5 gives a visual example of iterating a $3$-rule map over $X_{3}$.

Figure 5. Demonstration of six iterations of a $3$-rule map $T_{3}$ over
$X_{3}$, where $T_{3}(x_{0})=x_{1}$, and $T_{3}^{6}(x_{0})=x_{6}$. The
defining rule sequence of $T_{3}$ is $(r_{1}(x;\pi/2,L_{2})$,
$r_{2}(x;\theta_{2},1,L_{1})$, $r_{3}(x;\theta_{3},0,L_{3}))$. The solid lines
correspond to rule $r_{1}$, the dashed lines to $r_{2}$, and the dotted lines
to $r_{3}$.

Unless otherwise stated, the pair $(X_{m},T_{n})$ denotes a dynamical system.
For a point $x\in X_{m}$, we let $\mathcal{O}(x)$ denote the _orbit of $x$_
under $n$-rule map $T_{n}$, so that

$\mathcal{O}(x)\coloneqq\\{x,T_{n}(x),T_{n}^{2}(x),...\\}.$

We call an $n$-rule map _redundant_ if there exists a length $n^{\prime}$ rule
sequence with $m\leq n^{\prime}<n$, such that for all $x\in X_{m}$, the orbit
of $x$ under the $n^{\prime}$-rule map is equal to the orbit of $x$ under the
$n$-rule map. For the purpose of this paper we assume all $n$-rule maps are
not redundant.

###  2.1. n-Rule maps and rules as similarities

Let $L_{1},L_{2}$ denote lines in $\mathbb{R}^{2}$, intersecting at point $z$
with acute or right angle $\delta$. Let $x,y\in L_{1}$ lie on the same side of
the line intersection point $z$, and take a rule $r$ that projects onto line
$L_{2}$ with projection angle $\theta\in(0,\pi/2]$. Further, let the
orientation value of $r$ be chosen so that it maps $x$ and $y$ farthest from
$z$ when $\delta$ is acute (see Figure 4).

Let $d$ be the Euclidean metric. Assume $d(x,y)=\epsilon>0$, and $a=d(z,x)$,
$a+\epsilon=d(z,y)$. Let $\gamma=\pi-\delta-\theta$, so that $\gamma=\angle
zxr(x)=\angle zyr(y)$. Then it follows by use of the law of sines that

$d(r(x),r(y))=\left|\left|\frac{a\sin(\gamma)}{\sin(\theta)}-\frac{(a+\epsilon)\sin(\gamma)}{\sin(\theta)}\right|\right|_{2}=\frac{\epsilon\sin(\gamma)}{\sin(\theta)}.$

Let $c=\sin(\gamma)/\sin(\theta)$, and hence $d(r(x),r(y))=cd(x,y)$. We see
that if $0<\theta<(\pi-\delta)/2$ and $\delta$ is acute, then

$c=\frac{\sin(\gamma)}{\sin(\theta)}>1.$

Furthermore, if $\theta=(\pi-\delta)/2$, then $c=1$, and when

$\frac{\pi-\delta}{2}<\theta\leq\frac{\pi}{2}$

then $0\leq c<1$.

By similar argument, when the rule $r$ has opposite orientation parameter (and
hence maps $x$ and $y$ closer to $z$ in this case), we see that
$d(r(x),r(y))=cd(x,y)$ for some constant $c=c(\theta,\delta)$, computable
using the law of sines, where $0\leq c<1$ when $\delta/2<\theta\leq\pi/2$, and
$c=1$ when $\theta=\delta/2$, and $c>1$ when $0<\theta<\delta/2$.

When the points $x$ and $y$ are on opposite sides of the line intersection
point $z$, it still holds that $d(r(x),r(y))=cd(x,y)$. To see this, let
$x,y\in L_{1}$ lie on opposite sides of line intersection point $z$. Then
$d(r(x),z)=cd(x,z)$ and $d(r(y),z)=cd(y,z)$, and hence

$d(r(x),r(y))=d(r(x),z)+d(r(y),z)=cd(x,z)+cd(y,z)=cd(x,y)$

As such, by fixing the projection angle and orientation parameters $\theta$
and $o$ of a rule $r$, and restricting the mapping of a rule from one line to
another $X_{m}$, then $r$ becomes a similarity transformation. That is

$d(r(x),r(y))=cd(x,y),\text{ }c\geq 0.$

We call the constant $c$ a _similarity coefficient_.

Iteration of $n$-rule maps is defined to be a cycling composition of rules in
an associated rule sequence, and as a consequence, after the first iteration
of a $n$-rule map over a point in $X_{m}$, each rule in the rule sequence will
always map between the same pair of lines since each rule in the sequence
always projects onto the same line. Hence, by way of the above analysis,
iteration of a fixed $n$-rule map can be thought of as a cycling composition
of similarity transformations, after the first iteration of the map.

Let $\hat{T}_{n}\coloneqq T_{n}^{n}$ denote the induced map of $n$-rule map
$T_{n}$, so that $\hat{T}_{n}^{l}=T_{n}^{ln}$ for $l\in\mathbb{N}$, and
$\hat{T}_{n}:L_{a_{n}}\rightarrow L_{a_{n}}$, where $L_{a_{n}}$ is the line
the $n$th rule in the defining rule sequence of $T_{n}$ maps onto. If the
rules defining $T_{n}$ have similarity coefficients $c_{1},...,c_{n}$, then
let $C=c_{1}\cdot c_{2}\cdots c_{n}$ label the similarity coefficient for the
induced map $\hat{T}_{n}$.

##  3\. n-Rule maps and closed curves

In this section we prove Theorem 1.1. We begin by establishing the following
theorem.

Figure 6. Result from a numerical simulation of iterating a $6$-rule map in
$X_{4}$, with the orbit converging to a six-cycle.

######  Theorem 3.1.

Let $(X_{m},T_{n})$ be a dynamical system, and let $\hat{T}_{n}$ be the
induced map of $n$-rule map $T_{n}$, with similarity coefficient
$C=c_{1}c_{2}\cdots c_{n}$. If $0\leq C<1$, then $T_{n}$ admits a unique
periodic orbit of period $n$.

###### Proof.

Let $\hat{\mathcal{O}}(x)$ denote the orbit of $x\in X_{m}$ under
$\hat{T}_{n}$. It follows from definition of the induced map $\hat{T}_{n}$,
that for any $x\in X_{m}$, the orbit $\hat{\mathcal{O}}(x)\setminus\\{x\\}$ of
$x$ under $\hat{T}_{n}$, must be a subset of some line $L_{a_{n}}\subset
X_{m}$ determined by the $n$th rule in the defining rule sequence of $T_{n}$.
Further, by hypothesis $0\leq C<1$, and hence

$d(\hat{T}_{n}(x),\hat{T}_{n}(y))\leq Cd(x,y)$

for any $x,y\in L_{a_{n}}$, so $\hat{T}_{n}$ is a contraction mapping. But the
line $L_{a_{n}}$ is a closed subset of $\mathbb{R}^{2}$ and necessarily
complete. Hence, by the contraction mapping theorem there exists a unique
$x^{*}\in L_{i}$ such that $\hat{T}_{n}(x^{*})=x^{*}$. Then $T_{n}$ admits a
unique periodic orbit of period $n$. ∎

Refer to Figure 6 for visual demonstration of the type of dynamics Theorem 3.1
provides.

Assume two lines $L_{i},L_{j}$ in $X_{m}$ intersect at angle
$\delta\leq\pi/2$. Then if the defining rule sequence of an $n$-rule map
$T_{n}$ over $X_{m}$ contains a rule mapping from $L_{i}$ to $L_{j}$ (or vice-
versa) with projection angle $\theta=\delta$, then depending on rule
orientation, iteration of this rule may project onto the intersection point of
$L_{i}$ and $L_{j}$ every $n$ iterations, and hence the system collapses to a
periodic orbit after at most $n$ iterations of $T_{n}$. In such a case, we say
the $n$-rule map $T_{n}$ is _collapsing_. Collapsing maps correspond with the
case when the induced map of $n$-rule map $T_{n}$ has similarity coefficient
$C=0$. Figure 7 gives a visual example of a collapsing map.

######  Remark 1.

If an $n$-rule map is not collapsing, then it is invertible.

Figure 7. The closed curve corresponds to a periodic orbit generated by a
collapsing $5$-rule map. The map is composed of five rules, where
$\theta_{4}=\delta$, the acute line intersection angle of $L_{1}$ and $L_{2}$.
And hence because the fourth rule maps from $L_{1}$ to $L_{2}$, with
projection angle $\delta$ and orientation $1$, the intersection point of
$L_{1}$ and $L_{2}$ is mapped onto, becoming a periodic point.

If $m^{\prime}$ lines in $X_{m}$ intersect at a common point $z$, with $2\leq
m^{\prime}<m$, then a rule sequence $\\{r_{i}\\}_{i=1}^{n}$ may contain a
subsequence of consecutive rules which map strictly between the $m^{\prime}$
lines intersecting at $z$. As a consequence, such a subsequence of consecutive
rules would map $z$ to itself. In the case that such a line intersection point
$z$ is a periodic point, and a subsequence of rules maps over this point, we
say the periodic point $z$ is absorbing, and that the subsequence of rules is
an absorbed subsequence.

An absorbed subsequence may be composed of $1\leq k\leq n-2$ rules. That
$k\leq n-2$ is given by the nonconcurrency assumption of the lines composing
$X_{m}$, and that every line in $X_{m}$ must be mapped onto by at least one
rule in every rule sequence associated to an $X_{m}$. Hence, there are always
at least two rules in a rule sequence that cannot be absorbed.

We are now in a position to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Given a sequence $\theta_{1},\theta_{2},...,\theta_{n}$ of acute angles, as
well as a sequence $L_{a_{1}},L_{a_{2}},...,L_{a_{n}}$ of line labels over a
space $X_{m}$ with no two consecutive labels the same and each of the $m$
possible labels occuring at least once in the sequence, we may construct a
sequence of $n$ rules $\\{r_{i}\\}_{i=1}^{n}$ such that
$r_{i}=r(x;\theta_{i},o_{i},L_{a_{i}})$ with $i=1,...,n$ and leave each
$o_{i}$ arbitrary. Use this rule sequence to define an $n$-rule map $T_{n}$
over the fixed $X_{m}$, with the associated induced map $\hat{T}_{n}$ with
similarity coefficient $C=c_{1}c_{2}\cdots c_{n}$.

Consider the case when the angles $\theta_{1},...,\theta_{n}$ and other
parameters of the $n$-rule map are such that the similarity coefficient for
the induced map is $C<1$. Then by Theorem 3.1, $T_{n}$ admits a periodic
orbit, and by joining consecutive periodic points of this orbit with line
segments, we obtain a closed curve $\Gamma$ over $X_{m}$ that admits an
incidence angle sequence $\theta_{1},\theta_{2},...,\theta_{n}$ with respect
to the line sequence $L_{a_{1}},L_{a_{2}},...,L_{a_{n}}$, by the definition of
$T_{n}$. Similarly, if $T_{n}$ is collapsing, then it has a periodic orbit,
and this orbit corresponds to a closed curve $\Gamma$ admitting an incidence
angle sequence $\theta_{1},\theta_{2},...,\theta_{n}$ with respect to the line
sequence $L_{a_{1}},L_{a_{2}},...,L_{a_{n}}$.

If $C>1$ and $T_{n}$ is not collapsing, then $T_{n}$ has an inverse $n$-rule
map $T^{-1}_{n}$, with corresponding induced map $\hat{T}_{n}^{-1}$ and
similarity coefficient $C^{\prime}=1/C$. Then by Theorem 3.1 $T_{n}^{-1}$ has
a periodic orbit. But $T^{-1}_{n}$ is the inverse of $T_{n}$, and hence this
is also a periodic orbit for $T_{n}$. Constructing a closed cure $\Gamma$ from
this periodic orbit as above, we see that $\Gamma$ admits an incidence angle
sequence $\theta_{1},\theta_{2},...,\theta_{n}$ with respect to the line
sequence $L_{a_{1}},L_{a_{2}},...,L_{a_{n}}$.

Consider the case when $C=1$; we must show that $T_{n}$ still has a periodic
orbit. By the hypothesis of Theorem 1.1, every line defining $X_{m}$ must be
mapped onto at least once by $T_{n}$, so choose a rule $r_{i}$ in the defining
rule sequence that maps onto a line not parallel to or perpindicular with any
other line in $X_{m}$, which must exist by definition of $X_{m}$. Flip the
orientation of the rule $r_{i}$, so that if $o_{i}=0$, set $o_{i}=1$, and vice
versa. Then, since $r_{i}$ does not map between two parallel or perpindicular
lines, and has an acute projection angle by hypothesis of Theorem 1.1, it
follows that changing the orientation of the rule must change the value of $C$
so that $C$ no longer equals $1$. As such, we are left with either $C<1$ or
$C>1$, and we construct a closed curve by referring to the corresponding case
above.

Next, we note that when an $n$-rule map associated with a closed curve
$\Gamma$ is collapsing and has one or more absorbing periodic points, then one
or more subsequences of rules are absorbed over a line intersection point in
$X_{m}$, and as a consequence $\Gamma$ will be degenerate: not admitting a
full incidence angle sequence $\theta_{1},...,\theta_{n}$. To construct a
nondegenerate closed curve apply the following procedure: when an $n$-rule map
contains an absorbed subsequence, flip the orientation of the rule immediately
before the absorbed subsequence; this can be done because an absorbed
subsequence can have length at most $n-2$, and by hypothesis of Theorem 1.1
none of the projection angles are $\pi/2$. Then, at least one of the
subsequent rules in the rule sequence will be removed from the absorbed
subsequence. Continue this process until there are no more absorbed
subsequences.

This algorithm must terminate because there cannot be more than $\binom{m}{2}$
unique line intersection points, and hence no more than $\binom{m}{2}$
absorbed subsequences. Furthermore, there are $2^{n}$ distinct $n$-rule maps
associated with the distinct rule orientation configurations, and
$\binom{m}{2}<2^{n}$ when $3\leq m\leq n$, implying there exists at least one
rule orientation configuration such that the associated $n$-rule map has no
absorbed subsequences.

Figure 8 gives an example of three different periodic orbits in $X_{3}$
generated by three $5$-rule maps which only differ from one-another in rule
orientation combinations. ∎

Figure 8. The three closed curves correspond to those generated by three
$5$-rule maps, which only differ from one-another in rule orientation
parameters.

$n$-Rule maps lend themselves to studying periodic billiard orbits in polygons
in a natural way. If a closed curve generated by an $n$-rule map is strictly
contained within a polygon cut out by the intersecting lines composing
$X_{m}$, then there are cases in which the parameters of the $n$-rule map can
be adjusted so that the angles of incidence equal the angles of reflection,
and the closed curve corresponds with a periodic billiard orbit in the
polygon. Conditions that assert when this is possible can be determined,
however further discussion is out of the scope of this paper.

##  4\. n-Rule maps defined using piecewise continuous rules

This section begins the second part of our analysis, in which we redefine
$n$-rule maps so that rules map onto lines not on a basis of some fixed line
label, but rather on a basis of a distance between points and lines. Rules
then become piecewise continuous, and this redefinition introduces
discontinuities that complicate the dynamics. In this section we give basic
results concerning the redefined $n$-rule maps, and then in Section 5 we prove
Theorem 5.1, which shows when their orbits are asymptotically periodic.

This redefined dynamical system generalizes the system studied in [everett],
and shares characteristics with piecewise isometric dynamical systems.
Furthermore, the interest in this dynamical system is partially given by
potential applications, and the fact that such a simple geometric dynamical
system can lead to rich phenomena which provide characteristics analogous to a
number of well studied areas in dynamics, such as billiards.

###  4.1. Redefining n-rule maps

Let $Y_{m}\subset\mathbb{R}^{2}$ label the space of $m\geq 3$ pairwise
nonparallel, nonconcurrent line in $\mathbb{R}^{2}$. If $L_{i},L_{j}$ are two
distinct lines in $Y_{m}$ where $x\in L_{i}$, we define

$d(x,L_{j})\coloneqq\inf\\{d(x,y)|y\in L_{j}\\}$

to be the distance between point $x$ and line $L_{j}$, where $d$ is the
Euclidean metric and $d(x,L_{i})=0$.

For every point $x\in Y_{m}$, we construct the set

$D(x)=\\{d(x,L_{i})|1\leq i\leq m\\}$

and define the partially ordered set $\mathcal{D}(x)\coloneqq(D(x),\leq)$. We
let $l$ denote an index on $\mathcal{D}(x)$, so that $l=i$, $1\leq i\leq m$
corresponds with the $i$th farthest line from point $x$. Note that if there
exist $m^{\prime}$ lines in $Y_{m}$, $m^{\prime}<m$, that are all the same
distance from point $x$, then there are $m^{\prime}$ values of $l$ that do not
correspond with a unique distance value in $\mathcal{D}(x)$, and thus do not
correspond with a unique line in $Y_{m}$.

######  Definition 4.1.

Let $\theta\in(0,\pi/2]$, $o\in\\{0,1\\}$, and $l\in\\{2,3,...,m\\}$. We call
a mapping $r:Y_{m}\rightarrow Y_{m}$ a _piecewise rule_ , if $r(x)$ is an
angle $\theta$, orientation $o$ projection $x\in Y_{m}$ onto the $l$th
farthest line from $x$. If the index $l$ corresponds with a distance value in
$\mathcal{D}(x)$ that is not unique in $\mathcal{D}(x)$, put $r(x)=x$.

As before, we notate piecewise rules as $r(x;\theta,o,l)$ to make the
parameters of the rule explicit. Furthermore, for the rest of the paper, we
will refer to “piecewise rules” as “rules” for convenience, and when needed
refer to the rules used in Sections 2 and 3 as “symbolic rules.”

In the case when the index $l$ corresponds to a distance value in
$\mathcal{D}(x)$ that is not unique so that $r(x)=x$, then we say $x$ is an
_invariant_ point under rule $r$. Figure 3 provides visual demonstration of
the composition of two rules,

$r_{1}(x)\coloneqq r(x;\theta_{1},1,2)\text{ and }r_{2}(x)\coloneqq
r(x;\theta_{2},0,3)$

over a point $x_{0}\in Y_{3}$, so that $r_{1}(x_{0})=x_{1}$ and
$r_{2}(r_{1}(x_{0}))=x_{2}$. Intuitively, rule $r_{1}$ maps to the closest
line from a point $x$, and rule $r_{2}$ maps to the farthest line from a point
$x$ when applied to a space $Y_{3}$. We leave rule orientation to be defined
as previously, with the convention made visually explicit for this new class
of rules in Figures 3 and 4.

We leave rule sequences and $n$-rule maps defined as before, except for noting
that $n$-rule maps defined by piecewise rules may only have one rule in the
defining rule sequence, unlike those defined with symbolic rules which
required at least three. We let $K_{n}$ denote an $n$-rule map where the
defining rules in the rule sequence are piecewise rules. We may call such maps
piecewise $n$-rule maps for clarity, although for the remainder of this paper
we will only work with piecewise $n$-rule maps, and hence we refer to them
simply as “$n$-rule maps” when the context is clear.

For any piecewise $n$-rule map $K_{n}$, it is required that at least one of
the rules in the associated rule sequence has index value $l>2$; such a
restriction ensures the dynamics of a piecewise $n$-rule map are nontrivial.
If all rules of the rule sequence have $l$ index value of $l=2$, then each
iteration maps to the “closest” line, and orbits approach a line intersection
point of $Y_{m}$, failing to exhibit behavior of interest.

Unless otherwise stated, the pair $(Y_{m},K_{n})$ denotes a dynamical system.
If a point $x^{*}\in Y_{m}$ is invariant for $n^{\prime}<n$ of the rules in
the $n$-rule sequence defining $K_{n}$, then we say $x^{*}$ is _sometimes
invariant_ under $K_{n}$. If $x^{*}$ is invariant under all rules defining
$K_{n}$, we say $x^{*}$ is _strictly invariant_ under $K_{n}$. As such, any
1-rule map has only strictly invariant points.

######  Remark 2.

For any $(Y_{m},K_{n})$ dynamical system, the set of strictly invariant and
sometimes invariant points is finite.

As before, we call an $n$-rule map _redundant_ if there exists a length
$n^{\prime}$ rule sequence with $1\leq n^{\prime}<n$, such that for all $x\in
Y_{m}$, the orbit of $x$ under the $n^{\prime}$-rule map is equal to the orbit
of $x$ under the $n$-rule map. We assume all piecewise $n$-rule maps are not
redundant.

###  4.2. Convergence and contraction of piecewise n-rule maps

We now give results pertaining to piecewise $n$-rule maps that are used in
determining asymptotic behavior of orbits.

The space $Y_{m}$ is composed of $m$ pairwise nonparallel, nonconcurrent
lines, so any given space $Y_{m}$ has $\binom{m}{2}$ pairwise line
intersection points. Then, for each pairwise intersection point, let
$\eta_{i}$ label the $i$th pairwise line intersection angle, where
$0<\eta_{i}\leq\pi/2$. Let

$\delta=\min\left\\{\eta_{i}|1\leq i\leq\binom{m}{2}\right\\}$

label the least pairwise intersection angle between any two lines in $Y_{m}$.
Note $\delta$ must be acute by definition of $Y_{m}$.

######  Definition 4.2 (Average Contraction Condition).

For piecewise $n$-rule map $K_{n}$, let $\bar{\theta}$ label the average of
all projection angles in the $n$-rule sequence defining $K_{n}$. Then if

(1) $\frac{\pi-\delta}{2}<\bar{\theta}\leq\frac{\pi}{2}$

for least angle $\delta$ in $Y_{m}$, we say $K_{n}$ satisfies the _average
contraction condition_ with respect to $Y_{m}$.

We motivate the introduction of the average contraction condition through the
following observations, which are similar to those given in Section 2.1.

Let $L_{1},L_{2}$ denote lines in $\mathbb{R}^{2}$, intersecting at point $z$
with acute angle $\delta$. Without loss of generality, let $x,y\in L_{1}$, and
take a rule $r$, such that $r(x),r(y)\in L_{2}$, and $x,y,r(x),r(y)$ are on
the same side of intersection point $z$. Further, let the orientation value of
$r$ be the choice that maps farthest from $z$. For example, in Figure 4 rule
orientation value $o=1$ maps farther away from the line intersection point
when mapping from the particular line.

Assume $d(x,y)=\epsilon>0$, and $a=d(z,x)$, $a+\epsilon=d(z,y)$. Let $\theta$
denote the projection angle of rule $r$, and let $\gamma=\pi-\delta-\theta$,
so that $\gamma=\angle zxr(x)=\angle zyr(y)$. Then if

$0<\theta<\frac{\pi-\delta}{2}$

it follows by use of the law of sines that

$d(r(x),r(y))=\left|\left|\frac{a\sin(\gamma)}{\sin(\theta)}-\frac{(a+\epsilon)\sin(\gamma)}{\sin(\theta)}\right|\right|_{2}=\frac{\epsilon\sin(\gamma)}{\sin(\theta)}$

but $\theta<(\pi-\delta)/2$ and $\delta$ is acute, so under our choice of rule
orientation value

$\frac{\sin(\gamma)}{\sin(\theta)}>1$

and then $d(r(x),r(y))>d(x,y)$: iteration of $r$ over $L_{1}$ and $L_{2}$ in
such a way is then expansive. By similar argument, we see that if
$\theta=(\pi-\delta)/2$, then the rule defines an isometry and
$d(r(x),r(y))=d(x,y)$. When

$\frac{\pi-\delta}{2}<\theta\leq\frac{\pi}{2}$

then $d(r(x),r(y))\leq cd(x,y)$, $0\leq c<1$. Further, $\delta$ is acute, so
in the case when $r(x)$ maps opposite the angle $\pi-\delta$, we have
$\pi-\delta>\delta$, so if $r$ is contractive when mapping opposite $\delta$,
it must also be contractive when mapping opposite $\pi-\delta$.

From the above example, we see that for any rule $r$, with colinear $x,y$ and
colinear $r(x),r(y)$ all on the same side of the line intersection point, we
have

$d(r(x),r(y))\leq cd(x,y),\text{ }c\geq 0$

where $c$ can be computed directly via the law of sines, as a function of the
rule projection angle and the opposite line intersection angle. In this case,
we call such values $c$, _separation coefficients_.

######  Lemma 4.1.

Let lines $L_{1},L_{2}\subset Y_{m}$ intersect at a point $z$ with acute angle
$\delta$, and let $x,y\in L_{1}$ lie on the same side of $z$. Let
$r_{1},r_{2}$ label rules with distinct orientation values, which are chosen
so that the rules map farthest from $z$, and let the points
$r_{i}(x),r_{i}(y)\in L_{2}$ and $r_{i}(r_{j}(x)),r_{i}(r_{j}(y))\in L_{1}$,
$i,j=1,2$, $i\not=j$ all lie on the same side of $z$. Then for corresponding
rule separation constants $c_{1}$ and $c_{2}$, we have that $0\leq
c_{1}c_{2}<1$ if and only if

(2) $\frac{\pi-\delta}{2}<\frac{\theta_{1}+\theta_{2}}{2}\leq\frac{\pi}{2}$

for rule projection angles $\theta_{1},\theta_{2}$ corresponding with rules
$r_{1},r_{2}$.

Note that $c_{1}c_{2}<1$ implies composition of the two rules defines a
contraction:

$d(r_{i}(r_{j}(x)),r_{i}(r_{j}(y)))\leq c_{1}c_{2}d(x,y),\text{ }0\leq
c_{1}c_{2}<1$

for distinct $i,j$. Further, the orientation values of the rules are chosen so
that the rules map farthest from the line intersection point in each case, and
thus the corresponding separation constants are maximized.

###### Proof of Lemma 4.1.

Let $\gamma_{1}=\pi-\theta_{1}-\delta$ and $\gamma_{2}=\pi-\theta_{2}-\delta$.
We assume that

$\frac{\pi-\delta}{2}<\theta_{1}\leq\frac{\pi}{2}$

so by Equation 2, we require that

(3) $\pi-\theta_{1}-\delta<\theta_{2}\leq\pi-\theta_{1}$

By Equation 3 we see that $\sin(\theta_{2})>\sin(\pi-\theta_{1}-\delta)$, and
that

$\displaystyle\sin(\gamma_{2})$ $\displaystyle=\sin(\pi-\theta_{2}-\delta)$
$\displaystyle=\sin(\theta_{2}+\delta)$
$\displaystyle\leq\sin(\pi-\theta_{1}+\delta)$

Then, through substitution we obtain

$c_{1}c_{2}=\frac{\sin(\gamma_{1})}{\sin(\theta_{1})}\frac{\sin(\gamma_{2})}{\sin(\theta_{2})}<\frac{\sin(\pi-\theta_{1}-\delta)}{\sin(\theta_{1})}\frac{\sin(\pi-\theta_{1}+\delta)}{\sin(\pi-\theta_{1}-\delta)}$

but $\sin(\pi-\theta_{1}+\delta)=\sin(\theta_{1}-\delta)<\sin(\theta_{1})$, so

$\frac{\sin(\pi-\theta_{1}-\delta)}{\sin(\theta_{1})}\frac{\sin(\pi-\theta_{1}+\delta)}{\sin(\pi-\theta_{1}-\delta)}=\frac{\sin(\pi-\theta_{1}+\delta)}{\sin(\theta_{1})}<1$

Going the other direction, let $\gamma_{1}=\pi-\theta_{1}-\delta$ and
$\gamma_{2}=\pi-\theta_{2}-\delta$. Then from

$c_{1}c_{2}=\frac{\sin(\gamma_{1})\sin(\gamma_{2})}{\sin(\theta_{1})\sin(\theta_{2})}<1$

with substitution we obtain

$\sin(\pi-\theta_{1}-\delta)\sin(\pi-\theta_{2}-\delta)<\sin(\theta_{1})\sin(\theta_{2})$

By the product identity for sine, we have

$\frac{\cos(-\theta_{1}+\theta_{2})-\cos(2\pi-\theta_{1}-\theta_{2}-2\delta)}{2}<\frac{\cos(\theta_{1}-\theta_{2})-\cos(\theta_{1}+\theta_{2})}{2}$

but $\cos(-1(\theta_{1}-\theta_{2}))=\cos(\theta_{1}-\theta_{2})$ so upon
simplifying we have

$\cos(2\pi-\theta_{1}-\theta_{2}-2\delta)>\cos(\theta_{1}+\theta_{2})$

and by removing cosine we obtain

$\displaystyle 2\pi-\theta_{1}-\theta_{2}-2\delta$
$\displaystyle<\theta_{1}+\theta_{2}$

Note that although cosine is not monotone, by the restrictions on the angles
we can remove cosine in such a way. This gives us

$\pi-\delta<\theta_{1}+\theta_{2}\Longrightarrow\frac{\pi-\delta}{2}<\frac{\theta_{1}+\theta_{2}}{2}$

The case for the upper bound $(\theta_{1}+\theta_{2})/2\leq\pi/2$ is clear. ∎

In the above lemma, we took the rule orientation values to be chosen in a way
that ensures the rules map farthest from the line intersection point. If we
instead use rule orientation values that force the rules to map closer to the
line intersection points, then so long as the two projection angles
$\theta_{1},\theta_{2}$ satisfy Equation 2, both rules must provide
contraction (i.e. $c_{1}<1$ and $c_{2}<1$).

That is, if $\gamma_{2}=\pi-\theta_{2}-\delta$, and

$\frac{\sin(\gamma_{2})}{\sin(\theta_{2})}>1$

under a rule orientation value mapping farther from a line intersection point,
then under opposite rule orientation value, we have
$\gamma_{2}^{\prime}=\pi-(\pi-\theta_{2})-\delta$, so

$\frac{\sin(\gamma_{2}^{\prime})}{\sin(\pi-\theta_{2})}=\frac{\sin(\theta_{2}-\delta)}{\sin(\theta_{2})}<1.$

As an immediate consequence of the above remark and Lemma 4.1, we obtain the
following corollary.

######  Corollary 4.2.

Let $L_{1},L_{2}\subset Y_{m}$ intersect at point $z$ with acute angle
$\delta$. Further, let $K_{n}$ be a piecewise $n$-rule map so that iterates of
$K_{n}$ map between $L_{1}$ and $L_{2}$, opposite angle $\delta$, and for
initial points $x,y\in L_{1}$, let the first $n$ points of
$\mathcal{O}(x),\mathcal{O}(y)$ remain on the same side of $z$. Then if
$K_{n}$ satisfies the average contraction condition for least angle $\delta$,

$d(K^{n}_{n}(x),K^{n}_{n}(y))\leq Cd(x,y),\text{ }0\leq C<1.$

We remark that here $C=c_{1}c_{2}\cdots c_{n}$, is the product of the $n$
separation constants coming from the piecewise rule sequence.

######  Lemma 4.3.

For all lines $L_{i}\subset Y_{m}$ and $x,y\in L_{i}$, if each closed interval

$[K_{n}^{i}(x),K_{n}^{i}(y)]\subset Y_{m},\text{ }0\leq i\leq n$

contains no line intersection points or invariant points, then if $K_{n}$
satisfies the average contraction condition in $Y_{m}$,

$d(K^{n}_{n}(x),K^{n}_{n}(y))\leq Cd(x,y),\text{ }0\leq C<1.$

###### Proof.

Let $\delta$ label the least pairwise intersection angle in $Y_{m}$. Then by
Corollary 4.2, if $K_{n}$ satisfies the average contraction condition over
$Y_{m}$, and iteration of $K_{n}$ is strictly opposite angle $\delta$, then
$d(K^{n}_{n}(x),K^{n}_{n}(y))\leq Cd(x,y)$ for $C\in[0,1)$. But $\delta$ is
the least angle in $Y_{m}$, so if iteration of $K_{n}$ contracts opposite
angle $\delta$ on average, then it must also contract opposite every other
angle in $Y_{m}$ on average: if $\eta_{i}$ is a distinct line intersection
angle, then $\eta_{i}\geq\delta$, and

$\frac{\pi-\eta_{i}}{2}\leq\frac{\pi-\delta}{2}.$

As such, assuming the conditions of the statement, it follows that

$d(K^{n}_{n}(x)K^{n}_{n}(y))\leq Cd(x,y)$

for $C\in[0,1)$. ∎

We note the average contraction condition ensures contraction regardless of
rule orientation. The average contraction condition provides sufficient but
not necessary conditions for an $n$-rule map to define a contraction on
average.

######  Lemma 4.4.

If $K_{n}$ satisfies the average contraction condition over $Y_{m}$, then
there exists bounded regions $R,R^{\prime}\subset Y_{m}$ such that for all
$x\in R$, $\overline{\mathcal{O}(x)}\subset R^{\prime}$.

###### Proof.

By definition of $n$-rule maps and the average contraction condition,
iteration of an $n$-rule map $K_{n}$ in $Y_{m}$ must, on average, map closer
to line intersection points. The lines composing $Y_{m}$ are pairwise
nonparallel, so all lines must intersect, and there must exist a bounded
region $R$ containing all such line intersection points. As such, if iteration
of $K_{n}$ maps closer to line intersection points on average, then iteration
of the map must remain in a bounded region $R^{\prime}$. ∎

Immediate from proof of Lemma 4.4, we obtain the following corollary.

######  Corollary 4.5.

If $K_{n}$ satisfies the average contraction condition over $Y_{m}$, then any
sequence of points taken from successive preimages of $K_{n}$ over
noninvariant points $x\in Y_{m}$ diverges in $Y_{m}$.

##  5\. Asymptotic behavior of piecewise n-rule maps

In this section, we study the asymptotic properties of piecewise $n$-rule maps
satisfying the average contraction condition over $Y_{m}$. For piecewise
$n$-rule map $K_{n}$ and point $x\in Y_{m}$, we call a _cycle_ of $K_{n}$ over
$x$ the application of $K_{n}$ to $x$, $n$ times; the cycle of $x_{0}\in
Y_{m}$ under $K_{n}$ is the sequence of points $x_{0},x_{1},...,x_{n}$, where
$K_{n}^{n}(x_{0})=x_{n}$. We let $\mathcal{K}_{n}\coloneqq K_{n}^{n}$ label
the _cycle map_ of $K_{n}$, so that for $x_{0}\in Y_{m}$,
$\mathcal{K}_{n}(x_{0})=x_{n}$, and
$\mathcal{K}^{t}_{n}(x_{0})=K^{tn}_{n}(x_{0})=x_{tn}$.

If rule $r_{i}$ in the rule sequence of $n$-rule map $K_{n}$ has sometimes
invariant point $q$ in $Y_{m}$, then for every $h\in Y_{m}$ such that
$K^{i}_{n}(h)=q$ for $1\leq i\leq n$, we call $h$ a _pre-invariant_ point of
rule $r_{i}$. Associated with the $(Y_{m},K_{n})$ dynamical system, we let
$\Omega$ denote the set of invariant points of all types, as well as preimages
of the cycle map $\mathcal{K}_{n}$ from all pre-invariant points. Further, if
$p$ is a strictly invariant point under $K_{n}$, then all points $a\in Y_{m}$
such that $\mathcal{K}_{n}^{j}(a)=p$, $j\in\mathbb{Z}^{+}$, are also contained
in $\Omega$.

Put $Y^{\prime}_{m}=Y_{m}\setminus\Omega$. We call the dynamical system
$(Y_{m},K_{n})$ _degenerate_ when iteration of $K_{n}$ eventually maps to an
invariant point of any type; it follows that for the dynamical system
$(Y_{m}^{\prime},\mathcal{K}_{n})$ to be well defined, $(Y_{m},K_{n})$ must be
a non-degenerate dynamical system. Such degenerate systems arise at
bifurcation points, and the remainder of this section focuses on the study of
non-degenerate systems. The main result of this section is as follows.

######  Theorem 5.1.

Let $(Y_{m},K_{n})$ be a non-degenerate system, with piecewise $n$-rule map
$K_{n}$ satisfying the average contraction condition over $Y_{m}$. Then for
all $x\in Y_{m}^{\prime}$, the orbit $\mathcal{O}(x)$ converges to a periodic
orbit of period $kn$, $k\in\mathbb{Z}^{+}$.

Figure 9 illustrates the kind of dynamics Theorem 5.1 provides, showing the
periodic orbit iteration of a $4$-rule map converged to in a space $Y_{5}$.

Figure 9. Iteration of a piecewise $4$-rule map in $Y_{5}$, with the orbit
converging to a four-cycle. This was the output of a numerical simulation.

We need some preparatory lemmas to prove Theorem 5.1. First, note that as
consequence of Corollary 4.5, if $K_{n}$ satisfies the average contraction
condition, then sequences of points taken from preimages of invariant points
diverge in $Y_{m}$, so $\Omega$ is guaranteed not to be dense in $Y_{m}$ since
the collection of sometimes invariant and strictly invariant points is finite.
If, however, $K_{n}$ fails to satisfy the average contraction condition then
such a guarantee may not be made.

If $K_{n}$ satisfies the average contraction conditions in $Y_{m}$, then let
$U_{m}$ denote the set of open intervals $I_{a}\subset Y_{m}$ such that the
boundary values of each $I_{a}$ are given by elements in $\Omega$; no element
in $\Omega$ is contained within an open interval $I_{a}$. Let

$\hat{\mathcal{O}}(x)\coloneqq\\{x,\mathcal{K}_{n}(x),\mathcal{K}^{2}_{n}(x),...\\}$

denote the orbit of $x$ under cycle map $\mathcal{K}_{n}$.

######  Lemma 5.2.

For non-degenerate dynamical system $(Y_{m},K_{n})$ and piecewise $n$-rule map
$K_{n}$ satisfying the average contraction condition over $Y_{m}$, if
$I_{a}\in U_{m}$, then there exists an $I_{b}\in U_{m}$ such that
$\mathcal{K}_{n}[I_{a}]\subset I_{b}$, where $I_{a},I_{b}$ need not be
distinct.

###### Proof.

We proceed by contradiction and assume $\mathcal{K}_{n}[I_{a}]\subset
I_{b}\cup I_{c}$. By definition, the boundary values of each $I_{a}\in U_{m}$
are invariant points or preimages of invariant points under $\mathcal{K}_{n}$.
It follows that if $\mathcal{K}_{n}[I_{a}]\subset I_{b}\cup I_{c}$, then
$I_{a}=I_{d}\cup I_{e}$, as $I_{a}$ would contain preimages of such boundary
values, a contradiction. ∎

######  Lemma 5.3.

If $(Y_{m},K_{n})$ is non-degenerate with $K_{n}$ satisfying the average
contraction condition and $x\in Y^{\prime}_{m}$, then
$\hat{\mathcal{O}}(x)\subset\bigcup_{i=1}^{s}I_{i}$ for finite $s$ and
$I_{i}\in U_{m}$.

###### Proof.

We first note that taking $x\in Y_{m}^{\prime}$ is, by definition, equivalent
to taking $x\in I_{a}$, for $I_{a}$ in $U_{m}$. For any $(Y_{m},K_{n})$
dynamical system, there may only be a finite number of invariant points of any
type under $K_{n}$, and by Corollary 4.5, preimages of $n$-rule maps
satisfying the average contraction condition diverge from points in $Y_{m}$.
It then follows by definition of the set $\Omega$ and corresponding
construction of intervals in $U_{m}$, that for any bounded region $R\subset
Y_{m}$, there may only be a finite number of such intervals $I_{a}$ in $R$.
Further, by Lemma 4.4, orbits of $n$-rule maps satisfying the average
contraction condition must remain in a bounded region. Finally, by Lemma 5.2,
for every $I_{a}\in U_{m}$, $\mathcal{K}_{n}[I_{a}]\subset I_{b}$, and it thus
follows that the orbit of $x$ under $\mathcal{K}_{n}$ is contained in a finite
number of intervals. ∎

We call an interval $I_{c}\in U_{m}$ _confining_ if there is a
$t\in\mathbb{Z}^{+}$, $t=t(I_{c},K_{n})$, such that
$\mathcal{K}_{n}^{t}[I_{c}]\subset I_{c}$.

######  Lemma 5.4.

If $(Y_{m},K_{n})$ is a non-degenerate dynamical system with $n$-rule map
$K_{n}$ satisfying the average contraction conditions over $Y_{m}$, then there
exists a confining interval $I_{c}$ in $Y_{m}$, and iteration of
$\mathcal{K}_{n}$ over any $x\in Y_{m}^{\prime}$ maps into a confining
interval in a finite number of iterations.

###### Proof.

By Lemma 5.3, the orbit of any $x\in Y_{m}^{\prime}$ under $\mathcal{K}_{n}$
is restricted to a finite number of intervals. Thus, by way of the pigeon hole
principle, iteration of $\mathcal{K}_{n}$ is forced to map to an interval it
has already visited in a finite number of iterations: a confining interval.
And because the orbit is restricted to a finite number of intervals, it must
map into a confining interval after a finite number of iterations. ∎

######  Definition 5.1.

Let $I_{c}\in U_{m}$ be a confining interval in $Y_{m}$, and let
$\hat{\mathcal{K}}_{n}:I_{c}\rightarrow I_{c}$ be the _induced map_ of
$\mathcal{K}_{n}$ over the interval of continuity $I_{c}$, defined so that if
$x\in I_{c}$ and $\mathcal{K}_{n}^{k}(x)\in I_{c}$ for minimal $k$, we put
$\hat{\mathcal{K}}_{n}(x)=\mathcal{K}^{k}_{n}(x)=K^{kn}_{n}(x)$.

######  Lemma 5.5.

If $(Y_{m},K_{n})$ is a non-degenerate dynamical system with $n$-rule map
$K_{n}$ satisfying the average contraction condition over $Y_{m}$, and let
$I_{c}$ be a confining interval in $Y_{m}$. Then the induced map
$\hat{\mathcal{K}}_{n}$ has a unique fixed point in $I_{c}$.

###### Proof.

By hypothesis, $K_{n}$ satisfies the average contraction condition over
$Y_{m}$, so $\hat{\mathcal{K}}_{n}$ is a contraction mapping over confining
interval $I_{c}$ as consequence of Lemma 4.3. Further, we take the system
$(Y_{m},K_{n})$ to be non-degenerate, so by Lemma 5.2,
$\hat{\mathcal{K}}_{n}[I_{c}]\subset I_{c}$ (strict subset). As such, for any
$x\in I_{c}$, the sequence of points
$x,\hat{\mathcal{K}}_{n}(x),\hat{\mathcal{K}}^{2}_{n}(x),...$ is a Cauchy
sequence, and must converge to a unique point in the interval of continuity
$I_{c}$. It follows that there is a point $x^{*}\in I_{c}$ such that
$\hat{\mathcal{K}}_{n}(x^{*})=x^{*}$. ∎

We now prove Theorem 5.1.

###### Proof of Theorem 5.1.

By hypothesis, $(Y_{m},K_{n})$ is a non-degenerate dynamical system, $K_{n}$
satisfies the average contraction condition, and we take $x\in Y_{m}^{\prime}$
so iteration of $K_{n}$ over $x$ does not map to an invariant point of any
type. It then follows as a consequence of Lemma 5.4 that iteration of $K_{n}$
over $x\in Y_{m}^{\prime}$ maps into a confining interval $I_{c}\in U_{m}$ in
a finite number of iterations. And by consequence of Lemma 5.5 and Definition
5.1, iteration of $K_{n}$ in a confining interval must converge to a periodic
orbit of period $kn$, $k\in\mathbb{Z}^{+}$. ∎

We remark that for particular periodic orbits generated by an $n$-rule map in
$Y_{m}$, we cannot claim that the corresponding basin of attraction is all of
$Y^{\prime}$, as the periodic orbit is also dependent on initial condition
$x_{0}\in Y^{\prime}$. Indeed, work established in [nogueira] for example,
which concerns piecewise contractions of the interval, motivates questions
regarding upper bounds for the number of distinct periodic orbits a fixed
$(Y_{m},K_{n})$ dynamical system can admit. One other question that arises
from our analysis is whether there are conditions that can be used to tell
whether a dynamical system $(Y_{m},K_{m})$ is degenerate or not.

Software that can be used to simulate both types of $n$-rule maps is publicly
available at [Everett_N-Rule-Maps_2020].

## References