# Shadowing as a Structural Property of the Space of Dynamical Systems

Jonathan Meddaugh  Baylor University, Waco TX, 76798
<EMAIL_ADDRESS>
###### Abstract.

We demonstrate that there is a large class of compact metric spaces for which
the shadowing property can be characterized as a structural property of the
space of dynamical systems. We also demonstrate that, for this class of
spaces, in order to determine whether a system has shadowing, it is sufficient
to check that _continuously generated_ pseudo-orbits can be shadowed.

###### Key words and phrases:

shadowing, pseudo-orbit tracing property, topological dynamics

###### 2020 Mathematics Subject Classification:

37B65, 37B45

##  1\. Introduction

For a function $f:X\to X$ on a metric space $X$, a $\delta$-pseudo-orbit is a
sequence $\langle x_{i}\rangle$ in $X$ which satisfies
$d(f(x_{i}),x_{i+1})<\delta$. These approximations of orbits arise naturally
when orbits are computed numerically and can diverge significantly from the
orbits that they are approximating. However, there is a large class of
functions for which sufficiently precise pseudo-orbits can be tracked within a
given level of tolerance. Such systems are said to have the _shadowing
property_ (or _pseudo-orbit tracing property_). It is immediately clear that
this property is important in numerical modeling of dynamical systems,
especially for those in which errors are likely to grow (see [6, 7, 19, 20]).

While it has natural applications in the setting of numerical modeling,
shadowing has significant theoretical importance. Indeed, the origin of the
shadowing property begins with analysis of of Anosov and Axiom A
diffeomorphisms–Bowen implicitly used this property in his proof that the
nonwandering set of an Axiom A diffeomorphism is a factor of a subshift of
finite type [2]. Later, Sinai and Bowen demonstrated explicitly that the
shadowing property holds for large subsystems of these diffeomorphisms [3,
24]. Since then, it has been observed that shadowing plays an important role
in stability theory [21, 23, 25] and in characterizing $\omega$-limit sets [1,
2, 14]. Shadowing has also been shown to be a relatively common property in
the space of dynamical systems on certain classes of spaces [5, 11, 13, 15,
16, 22, 26].

In this paper, we demonstrate that, for a large class of spaces, the shadowing
property is a _structural property_ of the space of dynamical systems ($X^{X}$
or $\mathcal{C}(X)$), in that whether a system has shadowing can be observed
only by examining its neighborhood basis in that space. In particular, the
main results of this paper are the following theorems.

######  Theorem 7.

Let $X$ be a compact metric space and $f\in\mathcal{C}(X)$. The dynamical
system $(X,f)$ has shadowing if and only if for every $\epsilon>0$ there
exists $\delta>0$ such that if $g\in X^{X}$ with $\rho(g,f)<\delta$, then each
$g$-orbit is $\epsilon$-shadowed by an $f$-orbit.

######  Theorem 18.

Let $X$ be a compact metric space which is perturbable and let $f\in C(X)$.
The dynamical system $(X,f)$ has shadowing if and only if for every
$\epsilon>0$ there exists $\delta>0$ such that if $g\in\mathcal{C}(X)$ with
$\rho(g,f)<\delta$, then each $g$-orbit is $\epsilon$-shadowed by an
$f$-orbit.

The organization of this paper is as follows. In Section 2 we formalize the
definitions and notation that we will use for the remainder of the paper. In
Section 3, we focus on the space $X^{X}$ of not necessarily continuous
dynamical systems and develop the theory leading up to Theorem 7. We also
prove results for nonautonomous dynamical systems. Section 4 discusses
analogous results for the space $\mathcal{C}(X)$ and culminates in the proof
of Theorem 18. Finally, in Section 5 we discuss the results of Sections 3 and
4 from different angles. We develop variations of the shadowing property that
encapsulate these results and also demonstrate that shadowing can be viewed as
a weak form of continuity for the function which maps a dynamical system to
its set of orbits.

##  2\. Preliminaries

A _dynamical system_ is a pair $(X,f)$ consisting of a compact metric space
$X$ and a function $f:X\to X$. This function need not be continuous, and is
not assumed to be unless otherwise stated.

For a fixed space $X$, we can identify a dynamical system with the function
alone, allowing us to consider the space of dynamical systems on $X$. In
particular, let $X^{X}$ denote the space of functions from $X$ to $X$ with
topology generated by the supremum metric

$\rho(f,g)=\sup_{x\in X}d(f(x),g(x)).$

Also, let $\mathcal{C}(X)$ denote the subspace of $X^{X}$ consisting of the
continuous functions (maps) from $X$ to $X$.

For $x\in X$ and $f\in X^{X}$, the _orbit of $x$ under $f$_ (or, the $f$-orbit
of $x$) is the sequence $\langle f^{i}(x)\rangle_{i\in\omega}$ where $f^{0}$
denotes the identity. For $\delta>0$, a _$\delta$ -pseudo-orbit for $f$_ is a
sequence $\langle x_{i}\rangle_{i\in\omega}$ such that for all $i\in\omega$,
$d(f(x_{i}),x_{i+1})<\delta$. For $\epsilon>0$, we say that a sequence
$\langle x_{i}\rangle_{i\in\omega}$ in $X$ _$\epsilon$ -shadows_ a sequence
$\langle y_{i}\rangle_{i\in\omega}$ in $X$ provided that
$d(x_{i},y_{i})<\epsilon$ for all $i\in\omega$.

A dynamical system $(X,f)$ has _shadowing_ (or the _pseudo-orbit tracing
property_ , sometimes denoted _POTP_) provided that for all $\epsilon>0$,
there exists $\delta>0$ such that if $\langle x_{i}\rangle$ is a
$\delta$-pseudo-orbit for $f$, then there exists $z\in X$ such that for all
$i$, $d(x_{i},f^{i}(z))<\epsilon$, i.e. the orbit of $z$ _$\epsilon$ -shadows_
the pseudo-orbit $\langle x_{i}\rangle$.

For the purposes of this paper, it is useful to consider dynamical systems
consisting of _sequences_ of functions from $X$ to $X$. When a distinction is
necessary, we will refer to dynamical systems consisting of a single function
as _autonomous_ and those consisting of several functions as _nonautonomous_.
We will endow the space, $(X^{X})^{\omega}$, of such systems with the uniform
topology, with metric given by

$\rho(\langle f_{i}\rangle,\langle
g_{i}\rangle)=\sup_{i\in\omega}\rho(f_{i},g_{i}).$

Let $\langle f_{i}\rangle\in(X^{X})^{\omega}$. For $x\in X$, the _orbit of $x$
under $\langle f_{i}\rangle$_ (or, the $\langle f_{i}\rangle$-orbit of $x$) is
the sequence $\langle f^{i}_{0}(x)\rangle$ where $f^{i}_{0}$ denotes the
composition of the first $i$-many functions in the sequence, i.e.
$f^{i}_{0}=f_{i-1}\circ f_{i-2}\circ\cdots\circ f_{1}\circ f_{0}$ and
$f^{0}_{0}$ denotes the identity. For $\delta>0$, a _$\delta$ -pseudo-orbit
for $\langle f_{i}\rangle$_ is a sequence $\langle x_{i}\rangle$ such that for
all $i\in\omega$, $d(f_{i}(x_{i}),x_{i+1})<\delta$.

We say that a nonautonomous dynamical system $(X,\langle f_{i}\rangle)$ has
_shadowing_ provided that for all that for all $\epsilon>0$, there exists
$\delta>0$ such that if $\langle x_{i}\rangle$ is a $\delta$-pseudo-orbit for
$\langle f_{i}\rangle$, then there exists $z\in X$ such that for all $i$,
$d(x_{i},f^{i}_{0}(z))<\epsilon$, i.e. the orbit of $z$ _$\epsilon$ -shadows_
the pseudo-orbit $\langle x_{i}\rangle$.

It is worth noting that the autonomous dynamical system $(X,f)$ can be
naturally identified with the dynamical system $(X,\langle f\rangle)$ since it
is clear that the dynamical properties of the systems $(X,f)$ and $(X,\langle
f\rangle)$ are identical.

##  3\. Shadowing as a structural property in $X^{X}$

In this section, we examine shadowing as a structural property of dynamical
systems in the _space_ of (not necessarily continuous) dynamical systems. We
begin by developing a characterization of the shadowing property in the space
$(X^{X})^{\omega}$ of nonautonomous dynamical systems on $X$.

As a first step, we make the following observation:

######  Remark 1.

If $\langle f_{i}\rangle,\langle g_{i}\rangle\in(X^{X})^{\omega}$ with
$\rho(\langle f_{i}\rangle,\langle g_{i}\rangle)<\delta$, then every $\langle
g_{i}\rangle$-orbit is a $\delta$-pseudo-orbit for $\langle f_{i}\rangle$.

###### Proof.

Fix $\langle f_{i}\rangle,\langle g_{i}\rangle$ with $\rho(\langle
f_{i}\rangle,\langle g_{i}\rangle)<\delta$ and let $x\in X$. Then $\langle
g^{i}_{0}(x)\rangle$ satisfies

$d(f_{i}(g^{i}_{0}(x)),g^{i+1}_{0}(x))=d(f(g^{i}_{0}(x)),g_{i}(g^{i}_{0}(x)))\leq\rho(f_{i},g_{i})\leq\rho(\langle
f_{i}\rangle,\langle g_{i}\rangle)<\delta$

and is, therefore, a $\delta$-pseudo-orbit for $\langle f_{i}\rangle$. ∎

In other words, for a dynamical system, orbits of nearby systems are pseudo-
orbits for the original system. As we see in the following lemma, if $\langle
f_{i}\rangle\in(X^{X})^{\omega}$, the converse is true as well.

######  Lemma 2.

Let $\langle f_{i}\rangle\in(X^{X})^{\omega}$, $\delta>0$, and $\langle
x_{i}\rangle$ be a $\delta$-pseudo-orbit for $\langle f_{i}\rangle$. Then
there exists $\langle g_{i}\rangle\in(X^{X})^{\omega}$ such that $\rho(\langle
f_{i}\rangle,\langle g_{i}\rangle)\leq\delta$ and $\langle x_{i}\rangle$ is an
orbit for $\langle g_{i}\rangle$.

###### Proof.

For each $i\in\omega$, define $g_{i}$ as follows: $g_{i}(y)=f_{i}(y)$ for all
$y\in X\setminus\\{x_{i}\\}$ and $g_{i}(x_{i})=x_{i+1}$. By construction, the
sequence $\langle x_{i}\rangle$ is an orbit of $\langle g_{i}\rangle$.

Finally, it is easily seen that for each $i\in\omega$,
$\rho(f_{i},g_{i})=d(x_{i+1},f_{i}(x_{i}))<\delta$, and thus $\rho(\langle
f_{i}\rangle,\langle g_{i}\rangle)\leq\delta$. ∎

Since pseudo-orbits are identifiable with orbits for nearby systems in this
setting, we can easily verify that the shadowing property can be characterized
as follows.

######  Theorem 3.

Let $X$ be a metric space and $\langle f_{i}\rangle\in(X^{X})^{\omega}$. The
dynamical system $(X,\langle f_{i}\rangle)$ has shadowing if and only if for
all $\epsilon>0$, there exists $\delta>0$ such that if $\langle
g_{i}\rangle\in(X^{X})^{\omega}$ with $\rho(\langle g_{i}\rangle,\langle
f_{i}\rangle)<\delta$, then each $\langle g_{i}\rangle$-orbit is
$\epsilon$-shadowed by an $\langle f_{i}\rangle$-orbit.

###### Proof.

First, suppose that the dynamical system $(X,\langle f_{i}\rangle)$ has
shadowing. Fix $\epsilon>0$ and choose $\delta>0$ to witness shadowing with
respect to $\epsilon$. If $\langle g_{i}\rangle$ is a sequence with
$\rho(\langle g_{i}\rangle,\langle f_{i}\rangle)<\delta$, then every $\langle
g_{i}\rangle$-orbit is a $\delta$-pseudo-orbit for $\langle f_{i}\rangle$, and
is thus shadowed by some $\langle f_{i}\rangle$-orbit.

Now, suppose that $(\langle f_{i}\rangle,X)$ has the indicated property. Fix
$\epsilon>0$ and choose $\delta_{0}>0$ to witness. Define
$\delta=\delta_{0}/2$ and let $\langle x_{i}\rangle$ be a $\delta$-pseudo-
orbit for $\langle f_{i}\rangle$. Then, by Lemma 2, there exists $\langle
g_{i}\rangle$ with $\rho(\langle f_{i}\rangle,\langle
g_{i}\rangle)\leq\delta<\delta_{0}$ such that $\langle x_{i}\rangle$ is a
$\langle g_{i}\rangle$-orbit, and as such, is $\epsilon$-shadowed by some
$\langle f_{i}\rangle$-orbit. ∎

Since autonomous systems can be viewed as members of $(X^{X})^{\omega}$, as an
immediate corollary, we have the following characterization of shadowing for
autonomous systems.

######  Corollary 4.

Let $X$ be a metric space and $f\in X^{X}$. The dynamical system $(X,f)$ has
shadowing if and only if for all $\epsilon>0$, there exists $\delta>0$ such
that if $\langle g_{i}\rangle\in(X^{X})^{\omega}$ with $\rho(\langle
g_{i}\rangle,\langle f\rangle)<\delta$, then each $\langle g_{i}\rangle$-orbit
is $\epsilon$-shadowed by an $f$-orbit.

Of course, this characterization requires knowledge of an entire neighborhood
of $\langle f\rangle$ in $(X^{X})^{\omega}$. For an autonomous dynamical
system, a characterization requiring only information in $X^{X}$ would be far
preferable. The primary obstacle to such a characterization is that a non-
periodic pseudo-orbit which visits the same point twice cannot be realized as
an orbit of an autonomous system—in order to guarantee shadowing, such pseudo-
orbits need to be accounted for.

If we assume that our domain space contains no isolated points, we are able to
bypass this obstacle for continuous dynamical systems by perturbing such
pseudo-orbits. This will allow us to characterize shadowing as a structural
property for autonomous dynamical systems on spaces with no isolated points.

######  Lemma 5.

Let $f\in\mathcal{C}(X)$ and $\gamma>0$. Then if $\langle x_{i}\rangle$ a
$\gamma$-pseudo-orbit for $f$ for which $x_{i}=x_{j}$ implies that
$x_{i+1}=x_{j+1}$, then there exists $g\in X^{X}$ with $\rho(g,f)\leq\gamma$
and $z\in X$ such that for all $i\leq N$, $g^{i}(z)=x_{i}$.

###### Proof.

Fix $\gamma>0$ and let $\langle x_{i}\rangle$ be a $\gamma$-pseudo-orbit for
$f$ satisfying the hypotheses of the lemma.

Now, define $g:X\to X$ by

$g(x)=\begin{cases}x_{i+1}&x=x_{i}\\\
f(x)&x\notin\\{x_{i}:i\in\omega\\}\end{cases}.$

Note that $g$ is well-defined since if $x_{i}=x_{j}$, then
$g(x_{i})=x_{i+1}=x_{j+1}=g(y_{i})$.

By construction, $g(x_{i})=x_{i+1}$ and for all $x\in X$,
$d(f(x),g(x))\leq\sup\\{d(f(x_{i}),x_{i+1}):i\in\omega\\}\leq\gamma$, i.e.
$\rho(f,g)\leq\gamma$. ∎

######  Lemma 6.

Let $X$ be a compact metric space and let $f\in C(X)$ such that for every
$\epsilon>0$ there exists $\delta>0$ such that if $g\in X^{X}$ with
$\rho(g,f)<\delta$, then each $g$-orbit is $\epsilon$-shadowed by an
$f$-orbit. If $p\in X$ is isolated, then for every $\delta^{\prime}$ there
exists $\gamma_{p}>0$ such that if $\langle x_{i}\rangle$ is a
$\gamma_{p}$-pseudo-orbit for $f$, then either

  1. (1)

$|\\{i\in\omega:p=x_{i}\\}|\leq 1$ or

  2. (2)

$p$ is periodic and there exists $M\in\omega$ with $x_{M}$ in the orbit of $p$
and for any such $M$, $\sup\\{d({x_{M+i}},{f^{i}(x_{M})})\\}<\delta^{\prime}$.

###### Proof.

Fix $\delta^{\prime}>0$. Let $p\in X$ be isolated and suppose that for every
$\gamma>0$ there exists a $\gamma$-pseudo-orbit for $f$ in which $p$ appears
at least twice (else we can choose $\gamma_{p}>0$ such that condition (1)
holds for every $\gamma_{p}$-pseudo-orbit). Since $p$ is isolated, we can
choose $\epsilon>0$ with $B_{\epsilon}(p)=\\{p\\}$, and by supposition, there
is $\delta>0$ such that if $\rho(g,f)<\delta$, then every $g$-orbit is
$\epsilon$-shadowed by an $f$-orbit. Finally, choose $\delta>\gamma>0$.

By supposition, we can find a $\gamma$-pseudo-orbit $\langle x_{i}\rangle$ for
$f$ in which $p$ appears at least twice—let $M\in\omega$ and $N>0$ be
minimally chosen so that $p=x_{M}=x_{N+M}$.

Define, for $i<N$, $y_{i}=x_{M+i}$ and consider the finite sequence
$y_{0},\ldots,y_{N-1}$. Notice that $y_{0}=p$, $d(f(y_{N-1}),y_{0})<\gamma$,
and for all $i<N-1$, we have $d(f(y_{i}),y_{i+1})<\gamma$. Suppose that there
is an element of $X$ that occurs more than once in this list. Choose $n<N-1$
and $t>0$ such that $y_{n}=y_{n+t}$ and define the finite sequence
$y^{\prime}_{0},\ldots y^{\prime}_{N^{\prime}-1}$ as follows. Define
$N^{\prime}=N-t$ and

$y^{\prime}_{i}=\begin{cases}y_{i}&i\leq n\\\ y_{i+t}&i>n\end{cases}$

and observe that $y^{\prime}_{0}=p$, $d(f(y^{\prime}_{N-1}),z_{0})<\gamma$,
and for all $i<N^{\prime}-1$, we have
$d(f(y^{\prime}_{i}),y^{\prime}_{i+1})<\gamma$. By iterating this process as
necessary, we arrive at a finite sequence $z_{0},\ldots z_{P-1}$ with no
repeated elements, such that $z_{0}=p$, $d(f(z_{P-1}),z_{0})<\gamma$, and for
$i<P-1$, $d(f(z_{i}),z_{i+1})<\gamma$. It is easy to see then that if we
define $z_{i}=z_{i\mod P}$ for $i\geq P$, $\langle z_{i}\rangle$ is a
$\gamma$-pseudo-orbit with $z_{kP}=p$ for all $k\in\omega$ which satisfies the
hypotheses of Lemma 5.

As such, we can find a function $g\in X^{X}$ with $\rho(g,f)\leq\gamma<\delta$
such that $\langle z_{i}\rangle$ is a $g$-orbit, and therefore is
$\epsilon$-shadowed by the $f$-orbit of some point. But that point and its
image under $f^{P}$ must lie in $B_{\epsilon}(p)=\\{p\\}$, and therefore the
shadowing point is $p$ itself and $p$ has period $P$.

Now, by uniform continuity of $f,f^{2},\ldots f^{P-1}$, choose $\gamma_{p}>0$
such that $\gamma_{p}<\min\\{\epsilon,\delta^{\prime}/2\\}/P$ and if
$d(a,b)<\gamma_{p}$, then
$d(f^{i}(a),f^{i}(b))<\min\\{\epsilon,\delta^{\prime}/2\\}/P$ for all $i<P$.
Now, let $\langle x_{i}\rangle$ be a $\gamma_{p}$-pseudo-orbit for $f$ with
$x_{M}$ in the orbit of $p$ with $k<P$ chosen such that $f^{k}(x_{M})=p$.
Then, for $i\leq P$, we have

$\displaystyle d(x_{M+i},f^{i}(x_{M}))$
$\displaystyle\leq\sum_{j=0}^{i-1}d\left(f^{(i-j)}(x_{M+j}),f^{(i-j-1)}(x_{M+j+1})\right)$
$\displaystyle=d(f(x_{M+i-1}),x_{M+i})+\sum_{j=0}^{i-2}d\left(f^{(i-j)}(f(x_{M+j})),f^{(i-j)}(x_{M+j+1})\right)$
$\displaystyle<\gamma_{p}+\sum_{j=0}^{i-2}\min\\{\epsilon,\delta^{\prime}/2\\}/P<\min\\{\epsilon,\delta^{\prime}/2\\}$

In particular, $x_{M+k}\in B_{\epsilon}(f^{k}(x_{M}))=B_{\epsilon}(p)=\\{p\\}$
and by induction (applying the above argument to $x_{M+k}=p$), we can
establish that $d(x_{M+k+i},f^{i}(p))<\delta^{\prime}/2$ for all $i\in\omega$,
and thus
$\sup\\{d({x_{M+i}},f^{i}(p))\\}\leq\delta^{\prime}/2<\delta^{\prime}$ as
desired.

∎

Lemmas 5 and 6 allow us to prove the following characterization of shadowing.

######  Theorem 7.

Let $X$ be a compact metric space and $f\in\mathcal{C}(X)$. The dynamical
system $(X,f)$ has shadowing if and only if for every $\epsilon>0$ there
exists $\delta>0$ such that if $g\in X^{X}$ with $\rho(g,f)<\delta$, then each
$g$-orbit is $\epsilon$-shadowed by an $f$-orbit.

###### Proof.

As with Theorem 3, if $(X,f)$ has shadowing, then the conclusion follows
easily.

So, let us now assume that $(X,f)$ has the indicated property and fix
$\epsilon>0$. Choose $\epsilon/2>\delta>0$ to witness that if $g\in X^{X}$
with $\rho(f,g)<\delta$, then each $g$-orbit is $\epsilon/2$ shadowed by an
$f$-orbit. Finally, choose $\delta>\gamma>0$.

Now, by uniform continuity of $f$, fix $\gamma/3>\beta$ such that if
$d(a,b)<\beta$, then $d(f(a),f(b))<\gamma/3$, and define

$S=\\{x\in X:B_{\beta}(x)\textrm{ is infinite}\\}.$

Observe that $X\setminus S$ is finite—otherwise, it would have a limit point
$p$ and a point $q\in X\setminus S$ with $p\in B_{\beta}(q)$. But then
$B_{\beta}(q)$ is infinite, a contradiction. For each $p\in X\setminus S$, use
Lemma 6 to choose $\gamma_{p}>0$ such that either (1)
$|\\{i\in\omega:p=x_{i}\\}|\leq 1$ or (2) $p$ is periodic and there exists
$M\in\omega$ with $x_{M}$ in the orbit of $p$ and for any such $M$,
$\sup\\{d({x_{M+i}},{f^{i}(x_{M})})\\}<\gamma/3$.

Now, fix $\delta_{0}$ to be the lesser of $\beta$ and $\min\\{\gamma_{p}:p\in
X\setminus S\\}$. We claim that every $\delta_{0}$-pseudo-orbit is
$\epsilon$-shadowed by an orbit for $f$.

In order to verify this, let $\langle x_{i}\rangle$ be a $\delta_{0}$-pseudo-
orbit for $f$. We will now define a nearby $\gamma$-pseudo-orbit $\langle
y_{i}\rangle$ for which $y_{i}=y_{j}$ implies $y_{i+1}=y_{j+1}$ to which we
will apply Lemma 5 to achieve our claimed results.

There are two cases to consider.

Case 1: $\langle x_{i}\rangle$ has the property that if $i<j$ with
$x_{i}=x_{j}$, then $x_{i}\in S$. In this case, we proceed inductively,
defining $y_{0}=x_{0}$ and, once $y_{i}$ is defined for all $i<n$, we define
$y_{n}=x_{n}$ if $x_{n}\notin S$ and choose $y_{n}\in B_{\beta}(x_{n})\cap
S\setminus\\{y_{i}:i<n\\}$ otherwise. Clearly,
$\sup\\{d({y_{i}},{x_{i}})\\}\leq\beta$ and for $i\in\omega$,

$\displaystyle\@text@daccent{(}f(y_{i}),y_{i+1})$ $\displaystyle\leq
d(f(y_{i}),f(x_{i}))+d(f(x_{i}),x_{i+1})+d(x_{i+1},y_{i+1})$
$\displaystyle<\gamma/3+\delta_{0}+\beta<\gamma.$

i.e., $\langle y_{i}\rangle$ is a $\gamma$-pseudo-orbit. Furthermore, by
construction if $i\neq j$, then $y_{i}\neq y_{j}$.

Case 2: There exists $p\in X\setminus S$ and $m<n$ with $x_{n}=x_{m}=p$.
Choose $M$ minimal so that $x_{M}$ is in the orbit of some point $p\notin S$
and there exists $M\leq m<n$ with $x_{m}=x_{n}=p$. By choice of $\delta_{0}$
and the existence of $m,n$ we are in case (2) of Lemma 6, and so $x_{M}$ is
periodic and $\sup\\{d({x_{M+i}},f^{i}(x_{M}))\\}<\gamma/3$.

For $i<M$, we choose $y_{i}$ as in Case 1, with the additional condition that
$y_{i}$ not be in the orbit of $x_{M}$.

For $i\geq M$, define $y_{i}=f^{i-M}(x_{M})$. It is then clear that $\langle
y_{i}\rangle$ is a $\gamma$-pseudo-orbit (by the same argument above) and that
$\sup\\{d({x_{i}},{y_{i}})\\}\leq\max\\{\beta,\gamma/3\\}=\gamma/3$. By
construction we see that the hypotheses of Lemma 5 are met.

In either case, we have a $\gamma$-pseudo-orbit $\langle y_{i}\rangle$
satisfying the hypotheses of Lemma 5 such that
$\sup\\{d({x_{i}},{y_{i}})\\}\leq\gamma/3$.

By applying Lemma 5, we can find a function $g:X\to X$ with
$\rho(g_{N},f)\leq\gamma<\delta$ and $z\in X$ such that for all $i\in\omega$,
$g^{i}(z)=y_{i}$.

By our choice of $\delta$, there exists $q\in X$ such that the $f$-orbit of
$q$ $\epsilon/2$-shadows the $g$-orbit of $z$. Thus, for $i\in\omega$, we have

$d(f^{i}(q),x_{i})\leq
d(f^{i}(q),g^{i}(z))+d(g^{i}(z),y_{i})+d(y_{i},x_{i})<\epsilon/2+0+\gamma/3<\epsilon.$

In other words, the $\delta_{0}$-pseudo-orbit $\langle x_{i}\rangle$ is
$\epsilon$-shadowed by the $f$-orbit of $q$. Thus $f$ has shadowing as
claimed. ∎

##  4\. Shadowing as a structural property in $\mathcal{C}(X)$

In this section, we develop results analogous to those of Section 3 in the
space of _continuous_ dynamical systems. It is worth highlighting that the
results of that section depended heavily on freedom with which functions can
be modified if continuity is not under consideration. However, modifying
continuous functions appropriately is far more delicate. Broadly, if the
domain under consideration allows for continuous functions to be perturbed
with enough freedom, then we can demonstrate that shadowing is a structural
property of the space of continuous dynamical systems.

We begin by defining the notion of perturbability.

######  Definition 8.

A metric space $X$ is _perturbable_ provided that for all $\epsilon>0$, there
exists $\delta>0$ such that for every continuous function $f:X\to X$, every
finite subset $F=\\{x_{i}:i\leq N\\}\subseteq X$, and every function
$g_{0}:F\to X$ with $\rho(g_{0},f|_{F})<\delta$, there exists a continuous
function $g:X\to X$ with $\rho(f,g)<\epsilon$ and $g|_{F}=g_{0}$.

It is worth noting that the collection of perturbable spaces is quite
large—totally disconnected spaces are perturbable, for example. Spaces with a
degree of generalized local convexity are also perturbable.

######  Definition 9 ([10]).

A _c-structure_ on a topological space $X$ is a function $F$ from the set of
nonempty finite subsets of $X$ into the collection of nonempty contractible
subspaces of $X$ such that if $A\subseteq B$, then $F(A)\subseteq F(B)$.

A nonempty subset $E$ of $X$ is an _$F$ -set_ provided that $F(A)\subseteq E$
for all finite nonempty $A\subseteq E$.

A space $X$ with c-structure $F$ is an _m.c.-space_ provided that each point
has a neighborhood base consisting of $F$-sets.

In the above definition, the function $F$ essentially plays the role in an
m.c.-space that the convex hull plays in a locally convex space. Spaces which
are m.c.-spaces include manifolds, topological graphs, and many other classes
of continua.

The following extension theorem for m.c.-spaces is a generalization of
Dugundji’s extension theorem [9] and will be useful going forward.

######  Theorem 10 ([10]).

Let $X$ be a metric space, $A\subseteq X$ closed, $Y$ a m.c.-space with
c-structure $F$ and $f:A\to Y$ continuous. Then there exists a continuous
function $g:X\to Y$ which extends $f$ such that $g(X)\subseteq E$ for any
$F$-set $E$ in $Y$ which contains $f(A)$.

In particular, we apply this theorem to establish the following result.

######  Proposition 11.

If a compact metric space $X$ is totally disconnected or has a c-structure
which makes it an m.c.-space, then $X$ is perturbable.

###### Proof.

Fix $X$, $\epsilon>0$ and $F\subseteq X$ finite.

First, suppose that $X$ has c-structure $F$ which makes it an m.c.-space.
Since $X$ is compact and metric, we can choose $\delta<\epsilon/2$ such that
for every point $x\in X$, the set $B_{\delta}(x)$ is an $F$-set

Now, fix $f:X\to X$ continuous and choose $g_{0}:F\to X$ a function with
$\rho(g_{0},f|F)<\delta$. Since $f$ is continuous on the compact metric space
$X$, choose $\eta>0$ such that if $d(a,b)<\eta$, then $d(f(a),f(b))<\delta$
and such that if $p,p^{\prime}\in F$ with $p\neq p^{\prime}$,
$d(p,p^{\prime})>\eta$.

Now, for each $p\in F$, let $X_{p}=\\{x\in X:d(x,p)\leq\eta/2\\}$ and let
$C_{p}=\partial X_{p}\cup\\{p\\}$. Define $g_{p}:C_{p}\to X$ by

$h_{p}(x)=\begin{cases}f(x)&x\in\partial X_{p}\\\ g_{0}(p)&x=p\end{cases}.$

By construction, for each $x\in C_{p}$, $d(x,p)<\eta$ and hence
$d(f(x),f(p))<\delta$. Since $g_{0}(p)$ is also within $\delta$ of $f(p)$, we
see that $h_{p}(C_{p})\subseteq B_{\delta}(f(p))$. Since the latter is convex,
and $C_{p}$ is a closed subset of $X_{p}$, by Theorem 2 of [10], since
$B_{\delta}(f(p))$ is an $F$-set containing $g_{p}(C_{p})$, there is a
continuous extension $g_{p}:X_{p}\to X$ of $h_{p}$ with $g_{p}(X_{p})$ a
subset $B_{\delta}(f(p))$.

By choice of $\eta$, $\\{X_{p}:p\in F\\}$ is a collection of pairwise disjoint
closed sets, and thus we can define the desired function $g:X\to X$ as follows

$g(x)=\begin{cases}g_{p}(x):x\in X_{p}\\\ f(x):x\notin\bigcup
X_{p}.\end{cases}$

Clearly, for $p\in F$, $g(p)=g_{p}(p)=h_{p}(p)=g_{0}(p)$. To see that
$\rho(g,f)<\epsilon$, observe that if $x\notin\bigcup X_{p}$, then
$f(x)=g(x)$, and if $x\in X_{p}$, then $d(x,p)\leq\eta/2$ and hence
$d(f(x),f(p))<\delta$. But $g(x)=g_{p}(x)$ is contained in $B_{\delta}(f(p))$
by construction, and hence $d(f(x),g(x))<2\delta<\epsilon$.

If $X$ is instead totally disconnected, we proceed as follows. First, choose
$\delta<\epsilon$.

Now, fix $f:X\to X$ and $g_{0}:F\to X$ with $\rho(g_{0},f|F)<\delta$. By
uniform continuity, choose $\eta>0$ such that if $d(a,b)<\eta$, then
$d(f(a),f(b))<\epsilon-\delta$. Since $X$ is compact and totally disconnected,
for each $p\in F$, choose a neighborhood $X_{p}$ of $p$ with diameter less
than $\eta$ and with empty boundary and such that $\\{X_{p}:p\in F\\}$ is a
collection of pairwise disjoint sets. Now, we define the desired function
$g:X\to X$ by:

$g(x)=\begin{cases}g_{0}(x):x\in X_{p}\\\ f(x):x\notin\bigcup
X_{p}.\end{cases}$

By construction, $g$ is continuous, maps $p$ to $g_{0}(p)$ for all $p\in F$
and for $x\notin\bigcup X_{p}$, $d(f(x),g(x))=0<\epsilon$. For $x\in X_{p}$,
we see that $d(x,p)<\eta$ and hence $d((f(x),f(p))<\epsilon-\delta$. Since
$d(f(p),g(p))=d(f(p),g_{0}(p))<\delta$, it follows that
$d(f(x),g(x))<\epsilon$, and thus we see $\rho(f,g)<\epsilon$. ∎

For the purposes of establishing analogs to Theorem 3 and Corollary 4, a
weaker form of perturbability is sufficient.

######  Definition 12.

A metric space $X$ is _weakly perturbable_ provided that for all $\epsilon>0$,
there exists $\delta>0$ such that for every continuous function $f:X\to X$ and
every pair $(x,y)\in X\times X$ with $d(y,f(x))<\delta$, there exists a
continuous function $g:X\to X$ with $\rho(f,g)<\epsilon$ and $g(x)=y$.

It is immediately clear that every perturbable space is weakly perturbable. It
is not difficult to see that the converse statement is false—the product of
the interval and the Cantor set is weakly perturbable but not perturbable.

######  Lemma 13.

Suppose $X$ is weakly perturbable. Let $\langle
f_{i}\rangle\in\mathcal{C}(X)^{\omega}$ and $\epsilon>0$. Then there exists
$\delta>0$ such that if $\langle x_{i}\rangle$ is a $\delta$-pseudo-orbit for
$\langle f_{i}\rangle$, then there exists $\langle
g_{i}\rangle\in\mathcal{C}(X)^{\omega}$ such that $\rho(\langle
f_{i}\rangle,\langle g_{i}\rangle)\leq\epsilon$ and $\langle x_{i}\rangle$ is
an orbit for $\langle g_{i}\rangle$ .

###### Proof.

Fix $\epsilon>0$ and choose $\delta>0$ to witness the weak perturbability of
$X$. Let $\langle x_{i}\rangle$ be a $\delta$-pseudo-orbit for $\langle
f_{i}\rangle$.

For each $i\in\omega$, notice that $d(f_{i}(x_{i}),x_{i+1})<\delta$, and thus
we can apply weak perturbability to choose a function $g_{i}\in\mathcal{C}(X)$
with $g_{i}(x_{i})=x_{i+1}$ and $\rho(g_{i},f_{i})<\epsilon$. It is then
immediate that $\langle x_{i}\rangle$ is an orbit for $\langle g_{i}\rangle$
and $\rho(\langle f_{i}\rangle,\langle g_{i}\rangle)\leq\epsilon$. ∎

With this result, Theorem 14 and Corollary 15 follow by the same argumentation
as Theorem 3 and Corollary 4, respectively.

######  Theorem 14.

Let $X$ be a space which is weakly perturbable and let $\langle
f_{i}\rangle\in\mathcal{C}(X)^{\omega}$. The dynamical system $(\langle
f_{i}\rangle,X)$ has shadowing if and only if for all $\epsilon>0$, there
exists $\delta>0$ such that if $\langle
g_{i}\rangle\in\mathcal{C}(X)^{\omega}$ with $\rho(\langle
g_{i}\rangle,\langle f_{i}\rangle)<\delta$, then each $\langle
g_{i}\rangle$-orbit is $\epsilon$-shadowed by an $\langle f_{i}\rangle$-orbit.

######  Corollary 15.

Let $X$ be a space which is weakly perturbable and let $f\in\mathcal{C}(X)$.
The dynamical system $(X,{f})$ has shadowing if and only if for all
$\epsilon>0$, there exists $\delta>0$ such that if $\langle
g_{i}\rangle\in\mathcal{C}(X)^{\omega}$ with $\rho(\langle
g_{i}\rangle,\langle f\rangle)<\delta$, then each $\langle g_{i}\rangle$-orbit
is $\epsilon$-shadowed by an ${f}$-orbit.

In order to prove an analog for Theorem 7, it would be desirable to have an
analog for Lemma 5, i.e. the ability to generate a pseudo-orbit as an orbit of
a nearby function. In the previous section, this was easily achievable, as
continuity was not a concern. However, for this result, we would want the
nearby function to be continuous. In general, this would likely require being
able to perturb a function on an _infinite_ set, and (since we are requiring
continuity) hence also on any limit points. However, in the compact setting,
we will see that being able to perturb a function on finitely many points is
sufficient.

######  Lemma 16.

Suppose $X$ is perturbable . Let $f\in\mathcal{C}(X)$ and $\delta>0$. Then
there exists $\gamma>0$ such that if $\langle x_{i}\rangle$ a $\gamma$-pseudo-
orbit for $f$ for which $x_{i}=x_{j}$ implies that $x_{i+1}=x_{j+1}$, then for
every natural number $N$, there exists $g\in\mathcal{C}(X)$ with
$\rho(g,f)<\delta$ and $z\in X$ such that for all $i\leq N$, $g^{i}(z)=x_{i}$.

###### Proof.

Fix $\delta>0$ and let $\gamma>0$ witness the perturbability of $X$ and let
$\langle x_{i}\rangle$ be a $\gamma$-pseudo-orbit for $f$ satisfying the
hypotheses of the lemma.

Now, fix $N$ and define $F=\\{x_{i}:i<N\\}$ and $g_{0}:F\to X$ by
$g_{0}(x_{i})=x_{i+1}$. Note that $g_{0}$ is well-defined since if
$x_{i}=x_{j}$, then $g_{0}(x_{i})=x_{i+1}=x_{j+1}=g_{0}(y_{i})$. By
perturbability of $X$ and choice of $\gamma$, there exists a continuous
$g_{N}:X\to X$ such that $\rho(g_{N},f)<\delta$ and for $z=x_{0}$ we have
$g^{i}(z)=x_{i}$ for $i\leq N$, as claimed. ∎

The following lemma is very much a natural analog to Lemma 6 and the proof
differs only in that it appeals to Lemma 17 rather than Lemma 6. We include a
full proof for completeness.

######  Lemma 17.

Let $X$ be a perturbable compact metric space and let $f\in C(X)$ such that
for every $\epsilon>0$ there exists $\delta>0$ such that if
$g\in\mathcal{C}(X)$ with $\rho(g,f)<\delta$, then each $g$-orbit is
$\epsilon$-shadowed by an $f$-orbit. If $p\in X$ is isolated, then for every
$\delta^{\prime}$ there exists $\gamma_{p}>0$ such that if $\langle
x_{i}\rangle$ is a $\gamma_{p}$-pseudo-orbit for $f$, then either

  1. (1)

$|\\{i\in\omega:p=x_{i}\\}|\leq 1$ or

  2. (2)

$p$ is periodic and there exists $M\in\omega$ with $x_{M}$ in the orbit of $p$
and for any such $M$, $\sup\\{d({x_{M+i}},{f^{i}(x_{M})})\\}<\delta^{\prime}$.

###### Proof.

Fix $\delta^{\prime}>0$. Let $p\in X$ be isolated and suppose that for every
$\gamma>0$ there exists a $\gamma$-pseudo-orbit for $f$ in which $p$ appears
at least twice (else we can choose $\gamma_{p}>0$ such that condition (1)
holds for every $\gamma_{p}$-pseudo-orbit). Since $p$ is isolated, we can
choose $\epsilon>0$ with $B_{\epsilon}(p)=\\{p\\}$, and by supposition, there
is $\delta>0$ such that if $\rho(g,f)<\delta$, then every $g$-orbit is
$\epsilon$-shadowed by an $f$-orbit. Finally, choose $\delta>\gamma>0$
satisfying the conclusion of Lemma 16.

By supposition, we can find a $\gamma$-pseudo-orbit $\langle x_{i}\rangle$ for
$f$ in which $p$ appears at least twice—let $M\in\omega$ and $N>0$ be
minimally chosen so that $p=x_{M}=x_{N+M}$.

Define, for $i<N$, $y_{i}=x_{M+i}$ and consider the finite sequence
$y_{0},\ldots,y_{N-1}$. Notice that $y_{0}=p$, $d(f(y_{N-1}),y_{0})<\gamma$,
and for all $i<N-1$, we have $d(f(y_{i}),y_{i+1})<\gamma$. Suppose that there
is an element of $X$ that occurs more than once in this list. Choose $n<N-1$
and $t>0$ such that $y_{n}=y_{n+t}$ and define the finite sequence
$y^{\prime}_{0},\ldots y^{\prime}_{N^{\prime}-1}$ as follows. Define
$N^{\prime}=N-t$ and

$y^{\prime}_{i}=\begin{cases}y_{i}&i\leq n\\\ y_{i+t}&i>n\end{cases}$

and observe that $y^{\prime}_{0}=p$, $d(f(y^{\prime}_{N-1}),z_{0})<\gamma$,
and for all $i<N^{\prime}-1$, we have
$d(f(y^{\prime}_{i}),y^{\prime}_{i+1})<\gamma$. By iterating this process as
necessary, we arrive at a finite sequence $z_{0},\ldots z_{P-1}$ with no
repeated elements, such that $z_{0}=p$, $d(f(z_{P-1}),z_{0})<\gamma$, and for
$i<P-1$, $d(f(z_{i}),z_{i+1})<\gamma$. It is easy to see then that if we
define $z_{i}=z_{i\mod P}$ for $i\geq P$, $\langle z_{i}\rangle$ is a
$\gamma$-pseudo-orbit with $z_{kP}=p$ for all $k\in\omega$ which satisfies the
hypotheses of Lemma 16.

As such, we can find a function $g\in\mathcal{C}(X)$ with $\rho(g,f)<\delta$
such that $\langle z_{i}\rangle$ is a $g$-orbit, and therefore is
$\epsilon$-shadowed by the $f$-orbit of some point. But that point and its
image under $f^{P}$ must lie in $B_{\epsilon}(p)=\\{p\\}$, and therefore the
shadowing point is $p$ itself and $p$ has period $P$.

Now, by uniform continuity of $f,f^{2},\ldots f^{P-1}$, choose $\gamma_{p}>0$
such that $\gamma_{p}<\min\\{\epsilon,\delta^{\prime}/2\\}/P$ and if
$d(a,b)<\gamma_{p}$, then
$d(f^{i}(a),f^{i}(b))<\min\\{\epsilon,\delta^{\prime}/2\\}/P$ for all $i<P$.
Now, let $\langle x_{i}\rangle$ be a $\gamma_{p}$-pseudo-orbit for $f$ with
$x_{M}$ in the orbit of $p$ with $k<P$ chosen such that $f^{k}(x_{M})=p$.
Then, for $i\leq P$, we have

$\displaystyle d(x_{M+i},f^{i}(x_{M}))$
$\displaystyle\leq\sum_{j=0}^{i-1}d\left(f^{(i-j)}(x_{M+j}),f^{(i-j-1)}(x_{M+j+1})\right)$
$\displaystyle=d(f(x_{M+i-1}),x_{M+i})+\sum_{j=0}^{i-2}d\left(f^{(i-j)}(f(x_{M+j})),f^{(i-j)}(x_{M+j+1})\right)$
$\displaystyle<\gamma_{p}+\sum_{j=0}^{i-2}\min\\{\epsilon,\delta^{\prime}/2\\}/P<\min\\{\epsilon,\delta^{\prime}/2\\}$

In particular, $x_{M+k}\in B_{\epsilon}(f^{k}(x_{M}))=B_{\epsilon}(p)=\\{p\\}$
and by induction (applying the above argument to $x_{M+k}=p$), we can
establish that $d(x_{M+k+i},f^{i}(p))<\delta^{\prime}/2$ for all $i\in\omega$,
and thus $\sup\\{d({x_{M+i}},f^{i}(p))\\}<\delta^{\prime}$ as desired.

∎

With these tools in hand, we can now prove that, in perturbable spaces,
shadowing is a structural property of $\mathcal{C}(X)$. As with the previous
lemma, this theorem and its proof are very analogous to Theorem 7—again, we
include the full proof for completeness.

######  Theorem 18.

Let $X$ be a perturbable compact metric space and let $f\in C(X)$. The
dynamical system $(X,f)$ has shadowing if and only if for every $\epsilon>0$
there exists $\delta>0$ such that if $g\in\mathcal{C}(X)$ with
$\rho(g,f)<\delta$, then each $g$-orbit is $\epsilon$-shadowed by an
$f$-orbit.

###### Proof.

As in the proof of Theorem 3, if $(X,f)$ has shadowing, then the conclusion
follows immediately.

So, let us now assume that $(X,f)$ has the indicated property and fix
$\epsilon>0$. Choose $\epsilon/4>\delta>0$ to witness that if
$g\in\mathcal{C}(X)$ with $\rho(f,g)<\delta$, then each $g$-orbit is
$\epsilon/4$ shadowed by an $f$-orbit. Now, by Lemma 16, choose
$\delta>\gamma>0$ such that if $\langle x_{i}\rangle$ is a $\gamma$-pseudo-
orbit for $f$ for which $x_{i}=x_{j}$ implies $x_{i+1}=x_{j+1}$, then the
function $g$ in the conclusion of that lemma satisfies $\rho(g,f)<\delta$.

Now, by uniform continuity of $f$, fix $\gamma/3>\beta$ such that if
$d(a,b)<\beta$, then $d(f(a),f(b))<\gamma/3$, and define

$S=\\{x\in X:B_{\beta}(x)\textrm{ is infinite}\\}.$

Observe that $X\setminus S$ is finite—otherwise, it would have a limit point
$p$ and a point $q\in X\setminus S$ with $p\in B_{\beta}(q)$. But then
$B_{\beta}(q)$ is infinite, a contradiction. For each $p\in X\setminus S$, use
Lemma 17 to choose $\gamma_{p}>0$ such that either (1)
$|\\{i\in\omega:p=x_{i}\\}|\leq 1$ or (2) $p$ is periodic and there exists
$M\in\omega$ with $x_{M}$ in the orbit of $p$ and for any such $M$,
$\sup\\{d({x_{M+i}},{f^{i}(x_{M})})\\}<\gamma/3$.

Now, fix $\delta_{0}$ to be the lesser of $\beta$ and $\min\\{\gamma_{p}:p\in
X\setminus S\\}$ and let $\langle x_{i}\rangle$ be a $\delta_{0}$-pseudo-orbit
for $f$. We will now define a nearby $\gamma$-pseudo-orbit $\langle
y_{i}\rangle$ for which $y_{i}=y_{j}$ implies $y_{i+1}=y_{j+1}$ to which we
will apply Lemma 16 to achieve our claimed results.

There are two cases to consider.

Case 1: $\langle x_{i}\rangle$ has the property that if $i<j$ with
$x_{i}=x_{j}$, then $x_{i}\in S$. In this case, we proceed inductively,
defining $y_{0}=x_{0}$ and, once $y_{i}$ is defined for all $i<n$, we define
$y_{n}=x_{n}$ if $x_{n}\notin S$ and choose $y_{n}\in B_{\beta}(x_{n})\cap
S\setminus\\{y_{i}:i<n\\}$ otherwise. Clearly,
$\sup\\{d({y_{i}},{x_{i}})\\}\leq\beta$ and for $i\in\omega$,

$\displaystyle\@text@daccent{(}f(y_{i}),y_{i+1})$ $\displaystyle\leq
d(f(y_{i}),f(x_{i}))+d(f(x_{i}),x_{i+1})+d(x_{i+1},y_{i+1})$
$\displaystyle<\gamma/3+\delta_{0}+\beta<\gamma.$

i.e., $\langle y_{i}\rangle$ is a $\gamma$-pseudo-orbit. Furthermore, by
construction if $i\neq j$, then $y_{i}\neq y_{j}$.

Case 2: There exists $p\in X\setminus S$ and $m<n$ with $x_{n}=x_{m}=p$.
Choose $M$ minimal so that $x_{M}$ is in the orbit of some point $p\notin S$
and there exists $M\leq m<n$ with $x_{m}=x_{n}=p$. By choice of $\delta_{0}$
and the existence of $m,n$ we are in case (2) of Lemma 17, and so $x_{M}$ is
periodic and $\sup\\{d({x_{M+i}},f^{i}(x_{M}))\\}<\gamma/3$.

For $i<M$, we choose $y_{i}$ as in Case 1, with the additional condition that
$y_{i}$ not be in the orbit of $x_{M}$.

For $i\geq M$, define $y_{i}=f^{i-M}(x_{M})$. It is then clear that $\langle
y_{i}\rangle$ is a $\gamma$-pseudo-orbit (by the same argument above) and that
$\sup\\{d({x_{i}},{y_{i}})\\}\leq\max\\{\beta,\gamma/3\\}=\gamma/3$. By
construction we see that the hypotheses of Lemma 16 are met.

In either case, we have a $\gamma$-pseudo-orbit $\langle y_{i}\rangle$
satisfying the hypotheses of Lemma 16 such that
$\sup\\{d({x_{i}},{y_{i}})\\}\leq\gamma/3$.

By applying Lemma 16, for each $N$, we can fix a continuous $g_{N}:X\to X$
with $\rho(g_{N},f)<\delta$ and $z_{N}\in X$ such that for all $i\leq N$,
$g^{i}(z_{N})=y_{i}$.

By our choice of $\delta$, for each $N$, there exists $q_{N}\in X$ such that
the $f$-orbit of $q_{N}$ $\epsilon/4$-shadows the $g$-orbit of $z_{N}$. Thus,
for $i\leq N$, we have

$d(f^{i}(q_{N}),x_{i})\leq
d(f^{i}(q_{N}),g_{N}^{i}(z_{N}))+d(g_{N}^{i}(z_{N}),y_{i})+d(y_{i},x_{i})<\epsilon/4+0+\gamma/3<\epsilon/2.$

By passing to a subsequence if necessary, let $q=\lim_{N\to\infty}q_{N}$ and
observe that for all $i\in\omega$, we have
$d(f^{i}(q),x_{i})=\lim_{N\to\infty}d(f^{i}(q_{N}),x_{i})\leq\epsilon/2<\epsilon$,
i.e. the $\delta_{0}$-pseudo-orbit $\langle x_{i}\rangle$ is
$\epsilon$-shadowed by the $f$-orbit of $q$. Thus $f$ has shadowing as
claimed. ∎

Again, by Proposition 11, we have the following corollary.

######  Corollary 19.

Let $X$ be a compact metric space which is either totally disconnected or has
a c-structure which makes it an m.c.-space and let $f\in C(X)$. The dynamical
system $(X,f)$ has shadowing if and only if for every $\epsilon>0$ there
exists $\delta>0$ such that if $g\in\mathcal{C}(X)$ with $\rho(g,f)<\delta$,
then each $g$-orbit is $\epsilon$-shadowed by an $f$-orbit.

The notions of perturbability above are properties of the space under
consideration, however they can easily be rephrased as properties of
individual functions.

######  Definition 20.

A function $f:X\to X$ on a metric space $X$ is _perturbable_ provided that for
all $\epsilon>0$, there exists $\delta>0$ such that for every finite subset
$F=\\{x_{i}:i\leq N\\}\subseteq X$ and every function $g_{0}:F\to X$ with
$\rho(g_{0},f|_{F})<\delta$, there exists a continuous function $g:X\to X$
with $\rho(f,g)<\epsilon$ and $g|_{F}=g_{0}$.

Replacing the perturbability of $X$ with perturbability of $f$ in Lemma 16 and
Theorem 18 yields the following.

######  Theorem 21.

Let $X$ be a compact metric space and $f\in C(X)$ be perturbable. The
dynamical system $(X,f)$ has shadowing if and only if for every $\epsilon>0$
there exists $\delta>0$ such that if $g\in\mathcal{C}(X)$ with
$\rho(g,f)<\delta$, then each $g$-orbit is $\epsilon$-shadowed by an
$f$-orbit.

##  5\. Other Results

There has been significant recent interest in the study of variations of the
shadowing property [4, 12, 17, 18]. Broadly, many of these variations arise by
expanding or restricting the class of ‘pseudo-orbits’ that need to be shadowed
(or the degree to which they need to be shadowed). By Remark 1, if $(X,f)$ is
a dynamical system and $g$ is a nearby function, then the orbits of $g$ are
pseudo-orbits for $f$, and these pseudo-orbits are a proper subset of the full
collection of pseudo-orbits. This observation motivates the following
variations of the shadowing property.

######  Definition 22.

For $f\in X^{X}$ and $\delta>0$, a _functionally generated $\delta$-pseudo-
orbit_ is a sequence $\langle x_{i}\rangle$ such that there exists $g\in
X^{X}$ with $\rho(f,g)<\delta$ and $z\in X$ such that $x_{i}=g^{i}(z)$ for all
$i\in\omega$.

Functionally generated pseudo-orbits are a very natural type of pseudo-orbit
to consider. They arise naturally in finite precision computations of orbits,
for example. Notice that the functionally generated pseudo-orbits are a proper
subset of the collection of all pseudo-orbits, and thus the following notion
of shadowing is, _prima facie_ , weaker than the standard shadowing property.

######  Definition 23.

A dynamical system $(X,f)$ has the _finitely generated pseudo-orbit tracing
property (FGPOTP)_ provided that for all $\epsilon>0$ there exists $\delta>0$
such that every functionally generated $\delta$-pseudo-orbit $\langle
x_{i}\rangle$, there exists $z\in X$ such that the orbit of $z$ under $f$
$\epsilon$-shadows $\langle x_{i}\rangle$.

Of course, we can restrict the class of pseudo-orbits under consideration
further by requiring them to be generated by nearby continuous maps.

######  Definition 24.

For $f\in X^{X}$ and $\delta>0$, a _continuously generated $\delta$-pseudo-
orbit_ is a sequence $\langle x_{i}\rangle$ such that there exists
$g\in\mathcal{C}(X)$ with $\rho(f,g)<\delta$ and $z\in X$ such that
$x_{i}=g^{i}(z)$ for all $i\in\omega$.

Restricting our class of pseudo-orbits to the continuously generated ones
gives rise to the following even weaker notion of shadowing.

######  Definition 25.

A dynamical system $(X,f)$ has the _continuously generated pseudo-orbit
tracing property (CGPOTP)_ provided that for all $\epsilon>0$ there exists
$\delta>0$ such that every functionally generated $\delta$-pseudo-orbit
$\langle x_{i}\rangle$, there exists $z\in X$ such that the orbit of $z$ under
$f$ $\epsilon$-shadows $\langle x_{i}\rangle$.

With these notions in mind, Theorems 7 and Theorem 18 immediately give us the
following result.

######  Corollary 26.

Let $f\in\mathcal{C}(X)$. Then

  1. (1)

$f$ has shadowing if and only if it has FGPOTP.

  2. (2)

If $f$ has FGPOTP, then it has CGPOTP.

  3. (3)

If $X$ is perturbable and $f$ has CGPTOP, then it has shadowing and hence
FGPOTP.

Of course, it would be interesting to know how ‘sharp’ this result is.

######  Question 27.

Are there systems which have CGPOTP but not FGPOTP?

Another interesting viewpoint of the results from Section 4 comes about by
considering the _set_ of orbits generated by a dynamical system. Orbits for
$(X,f)$ belong to the set $X^{\omega}$, and if we wish to consider shadowing-
type properties, the natural metric on $X^{\omega}$ to consider is the
_uniform metric_ , i.e.

$\rho(\langle x_{i}\rangle,\langle
y_{i}\rangle)=\sup_{i\in\omega}\\{d(x_{i},y_{i})\\}.$

We can then define the function $\mathcal{O}:\mathcal{C}(X)\to X^{\omega}$
which maps a function to its set of orbits, i.e. $\mathcal{O}(f)=\\{\langle
f^{i}(x)\rangle:x\in X\\}.$

When considering set-valued functions between topological spaces, there are
several distinct generalizations of continuity. For the purposes of this
paper, we consider the notion of _upper Hausdorff semicontinuity_. It is worth
mentioning that for non-compact codomains, upper Hausdorff semicontinuity is
generally weaker than upper semicontinuity [8].

######  Definition 28.

A set-valued function $g:A\to B$ between metric spaces $(A,d_{A})$ and
$(B,d_{B})$ is _upper Hausdorff semicontinuous_ at $a\in A$ provided that for
all $\epsilon>0$ there exists $\delta>0$ such that if $d_{A}(a,x)<\delta$,
then $g(x)\subseteq B_{\epsilon}(g(a))$.

Using Theorem 18, we can prove the following result.

######  Theorem 29.

Let $X$ be a compact metric space which is perturbable and let $f\in C(X)$.
Then $f$ has shadowing if and only if $\mathcal{O}:\mathcal{C}(X)\to
X^{\omega}$ is upper Hausdorff semicontinuous at $f$.

###### Proof.

First, assume that $f$ has shadowing. Fix $\epsilon>0$ and let $\delta>0$
witness that every $\delta$-pseudo-orbit is $\epsilon/2$-shadowed. Observe
that if $\rho(f,g)<\delta$, and $\langle x_{i}\rangle\in\mathcal{O}(g)$, then
by Remark 1, $\langle x_{i}\rangle$ is a $\delta$-pseudo-orbit for $f$, and as
such, there exists $z\in X$ such that the $f$-orbit of $z$
$\epsilon/2$-shadows $\langle x_{i}\rangle$. But then $\rho(\langle
x_{i}\rangle,\langle f^{i}(z)\rangle)\leq\epsilon/2<\epsilon$, and since
$\langle f^{i}(z)\rangle\in\mathcal{O}(f)$, we have that $\langle
x_{i}\rangle\in B_{\epsilon}(\mathcal{O}(f))$. Thus $\mathcal{O}$ is upper
Hausdorff semicontinuous at $f$.

Now, suppose that $\mathcal{O}$ is upper Hausdorff semicontinuous at $f$. Fix
$\epsilon>0$ and choose $\delta>0$ so that if $\rho(f,g)<\delta$, then
$\mathcal{O}(g)\subseteq B_{\epsilon}(\mathcal{O}(f))$ . Let
$g\in\mathcal{C}(X)$ with $\rho(f,g)<\delta$ and let $x\in X$. Then $\langle
g^{i}(x)\rangle\in\mathcal{O}(g)\in B_{\epsilon}(\mathcal{O}(f))$. In
particular, there exists $z\in X$ such that $\rho(\langle
g^{i}(x)\rangle,\langle f^{i}(z)\rangle)<\epsilon$. In particular, for each
$i\in\omega$, $d(g^{i}(x),f^{i}(z))<\epsilon$, i.e. the $g$-orbit of $x$ is
$\epsilon$-shadowed by the $f$-orbit of $z$. Thus, by Theorem 18, $f$ has
shadowing. ∎

### Acknowledgements

The author would like to thank Professor Jim Wiseman for helpful feedback and
discussion in the revision of this paper.

## References

  * [1] A. D. Barwell, C. Good, P. Oprocha, and B. E. Raines.  Characterizations of $\omega$-limit sets of topologically hyperbolic spaces.  Discrete Contin. Dyn. Syst., 33(5):1819–1833, 2013.
  * [2] R. Bowen.  $\omega$-limit sets for axiom ${\rm A}$ diffeomorphisms.  J. Differential Equations, 18(2):333–339, 1975.
  * [3] Rufus Bowen.  Markov partitions for Axiom ${\rm A}$ diffeomorphisms.  Amer. J. Math., 92:725–747, 1970.
  * [4] William R. Brian, Jonathan Meddaugh, and Brian E. Raines.  Chain transitivity and variations of the shadowing property.  Ergodic Theory Dynam. Systems, 35(7):2044–2052, 2015.
  * [5] William R. Brian, Jonathan Meddaugh, and Brian E. Raines.  Shadowing is generic on dendrites.  Discrete Contin. Dyn. Syst., To Appear.
  * [6] R. M. Corless and S. Y. Pilyugin.  Approximate and real trajectories for generic dynamical systems.  J. Math. Anal. Appl., 189(2):409–423, 1995.
  * [7] Robert M. Corless.  Defect-controlled numerical methods and shadowing for chaotic differential equations.  Phys. D, 60(1-4):323–334, 1992.  Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991).
  * [8] Szymon Dolecki and Stefan Rolewicz.  Metric characterizations of upper semicontinuity.  J. Math. Anal. Appl., 69(1):146–152, 1979.
  * [9] James Dugundji.  Topology.  Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1978.  Reprinting of the 1966 original, Allyn and Bacon Series in Advanced Mathematics.
  * [10] Charles D. Horvath.  Extension and selection theorems in topological spaces with a generalized convexity structure.  Ann. Fac. Sci. Toulouse Math. (6), 2(2):253–269, 1993.
  * [11] Piotr Kościelniak, Marcin Mazur, Piotr Oprocha, and PawełPilarczyk.  Shadowing is generic—a continuous map case.  Discrete Contin. Dyn. Syst., 34(9):3591–3609, 2014.
  * [12] K. Lee and K. Sakai.  Various shadowing properties and their equivalence.  Discrete Contin. Dyn. Syst., 13(2):533–540, 2005.
  * [13] Jonathan Meddaugh.  On genericity of shadowing in one dimension.  Fund. Math., To Appear.
  * [14] Jonathan Meddaugh and Brian E. Raines.  Shadowing and internal chain transitivity.  Fund. Math., 222(3):279–287, 2013.
  * [15] Ivan Mizera.  Generic properties of one-dimensional dynamical systems.  In Ergodic theory and related topics, III (Güstrow, 1990), volume 1514 of Lecture Notes in Math., pages 163–173. Springer, Berlin, 1992.
  * [16] Kenzi Odani.  Generic homeomorphisms have the pseudo-orbit tracing property.  Proc. Amer. Math. Soc., 110(1):281–284, 1990.
  * [17] P. Oprocha.  Shadowing, thick sets and the ramsey property.  Ergodic Theory Dynam. Systems, 2015.
  * [18] Piotr Oprocha, Dawoud Ahmadi Dastjerdi, and Maryam Hosseini.  On partial shadowing of complete pseudo-orbits.  J. Math. Anal. Appl., 404(1):47–56, 2013.
  * [19] Ken Palmer.  Shadowing in dynamical systems, volume 501 of Mathematics and its Applications.  Kluwer Academic Publishers, Dordrecht, 2000.  Theory and applications.
  * [20] D. W. Pearson.  Shadowing and prediction of dynamical systems.  Math. Comput. Modelling, 34(7-8):813–820, 2001.
  * [21] S. Y. Pilyugin.  Shadowing in dynamical systems, volume 1706 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1999.
  * [22] S. Yu. Pilyugin and O. B. Plamenevskaya.  Shadowing is generic.  Topology Appl., 97(3):253–266, 1999.
  * [23] Clark Robinson.  Stability theorems and hyperbolicity in dynamical systems.  In Proceedings of the Regional Conference on the Application of Topological Methods in Differential Equations (Boulder, Colo., 1976), volume 7, pages 425–437, 1977.
  * [24] Ja. G. Sinaĭ.  Markov partitions and U-diffeomorphisms.  Funkcional. Anal. i Priložen, 2(1):64–89, 1968.
  * [25] P. Walters.  On the pseudo-orbit tracing property and its relationship to stability.  In The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), volume 668 of Lecture Notes in Math., pages 231–244. Springer, Berlin, 1978.
  * [26] Koichi Yano.  Generic homeomorphisms of $S^{1}$ have the pseudo-orbit tracing property.  J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34(1):51–55, 1987.