Continuity of ω-Limit Sets

• Continuity properties of ω-limit sets are defined via precise upper and lower semicontinuity criteria that capture asymptotic behavior in compact metric spaces.
• The analysis employs shadowing and chain continuity to establish robust equivalences and stability conditions in dynamical systems.
• Illustrative examples demonstrate how perturbations and topological constraints critically influence the persistence and minimality of ω-limit sets.

The continuity properties of $ω$-limit sets constitute a central topic in topological dynamics, describing the interplay between the asymptotic behavior of orbits, shadowing phenomena, and various notions of set-valued continuity. This exposition presents a rigorous synthesis of these concepts, their formal interrelations, and the explicit criteria under which upper and lower semicontinuity of $ω$-limit sets hold. The focus is on compact metric spaces $(X,d)$ under continuous maps $f:X\to X$, with precise attention to the nuanced roles played by shadowing and chain relations (Kawaguchi, 13 Jan 2026).

1. Preliminaries: $ω$-Limit Sets and Continuity Concepts

Given $(X,d)$ a compact metric space and $f:X\rightarrow X$ continuous, the $ω$-limit set of $x\in X$ is the collection of all possible accumulation points of its forward orbit: $$ω_f(x) = \left\{ y \in X :\, \exists\, 0 \leq i_1 < i_2 < \dots,\, f^{i_j}(x) \rightarrow y \right\}.$$ The $ω$-limit function $ω_f:X\rightarrow \mathcal{K}(X)$, where $\mathcal{K}(X)$ is the space of nonempty closed subsets with the Hausdorff metric $d_H$, is a set-valued map.

Upper semicontinuity (USC) and lower semicontinuity (LSC) for $ω_f$ at $x$ are formulated as:

• $ω_f$ is USC at $x$ if $\limsup_{y\to x} ω_f(y) \subseteq ω_f(x)$;
• $ω_f$ is LSC at $x$ if $ω_f(x) \subseteq \liminf_{y\to x} ω_f(y)$.

Here,

$$\limsup_{y\to x} ω_f(y)=\bigcap_{\delta>0}\bigcup_{d(x,y)<\delta}ω_f(y),\quad \liminf_{y\to x} ω_f(y)=\bigcup_{\delta>0}\bigcap_{d(x,y)<\delta}ω_f(y).$$

Define $USC(ω_f)$ and $LSC(ω_f)$ as the sets where $ω_f$ is upper or lower semicontinuous, and $C(ω_f)=USC(ω_f)\cap LSC(ω_f)$ the points of continuity.

2. Shadowing, Chain Structures, and Their Role

A sequence $(x_i)_{i\geq 0}$ is a δ-pseudo-orbit if $\sup_{i\geq 0} d(f(x_i),x_{i+1})\leq δ$. It is ε-shadowed by $y$ if $\sup_{i\geq 0} d(x_i, f^i(y))\leq ε$. If every δ-pseudo-orbit is ε-shadowed globally, $f$ has the global shadowing property. The set of shadowable points, $Sh(f)$, consists of those $x$ for which, for every $\epsilon>0$, some $\delta>0$ yields that every $\delta$-pseudo-orbit starting at $x$ is $\varepsilon$-shadowed.

Chain continuity at $x\in X$ means: for all $\epsilon>0$, some $\delta>0$ exists such that every $\delta$-pseudo-orbit with $x_0=x$ satisfies $\sup_{i\geq 0} d(x_i, f^i(x))\leq \epsilon$. The set $CC(f)$ collects all points of chain continuity. Chain recurrence is defined by $CR(f)=\{ x:\, x\to x \}$ (where $y\to z$ if for any δ, a δ-chain from $y$ to $z$ of arbitrary length exists). Equivalence classes of $CR(f)$ under $x\leftrightarrow y$ partition it into chain components $\mathcal{C}(f)$. A closed, $f$-invariant $C\subseteq X$ is chain stable if small chains initiated in $C$ remain close to $C$.

3. Criteria for Semi- and Continuity of $ω$-Limit Sets at Shadowable Points

3.1 Upper Semicontinuity

For $x\in Sh(f)$, the following conditions are equivalent:

• $x\in USC(ω_f)$;
• $ω_f(x)=Ω_f(x)$, where $$Ω_f(x)=\{ y :\, \exists\, x_j\to x \,\text{and}\, f^{i_j}(x_j)\to y \}$$.

This equivalence states that, at shadowable points, upper semicontinuity precisely captures the absence of new limit points arising from nearby initial conditions; that is, the $ω$-limit set is robust under small perturbations of initial data. The proof essentially combines the compactness of $X$, closed graph arguments, and exploits the shadowing property to show that any violation of equality would yield pseudo-orbits contradicting upper semicontinuity (Kawaguchi, 13 Jan 2026).

3.2 Lower Semicontinuity

For $x\in Sh(f)$, the following are equivalent:

• $x\in LSC(ω_f)$;
• $ω_f(x)\subseteq ω_f(y)$ for all $y\in Ω_f(x)$.

In particular, $ω_f(x)$ must be a minimal set under $(ii)$. The underlying argument leverages the shadowing property to prevent escape from the vicinity of $ω_f(x)$ by nearby orbits, ensuring the persistence of $ω$-limit set points under perturbations. Failure would indicate that shadowing can be used to construct a pseudo-orbit violating LSC.

3.3 Chain Stability and Minimality

From the above, for shadowable $x$, $ω_f(x)=Ω_f(x)$ if and only if $ω_f(x)$ is chain stable, and $x\in LSC(ω_f)$ implies minimality of $ω_f(x)$. As a corollary, at points where both USC and LSC hold and the point is shadowable, $ω_f(x)$ is both minimal and chain stable.

4. Global Shadowing, Chain Continuity, and Lower Semicontinuity

A central result is the equivalence, under global shadowing, of LSC of $ω$-limit sets and chain continuity: $X=LSC(ω_f)\quad \Longleftrightarrow \quad X=CC(f).$ For $f:X\to X$ continuous with $X=Sh(f)$, Theorem 5.1 shows that each $ω_f(x)$ is minimal and chain-stable under $LSC(ω_f)$. This minimality, combined with chain stability and the shadowing property, yields entropy zero systems, and terminal chain components are odometers or periodic orbits.

Conversely, if the system is chain continuous everywhere, equicontinuity ensures LSC everywhere. This equivalence highlights the deep structural ties between pseudo-orbit tracing (shadowing) and the robustness of $ω$-limit sets to perturbation.

For connected compact $X$, another characterization is given: $X=CC(f)$ if and only if the eventual image $\bigcap_{n\geq 0} f^n(X)$ is a singleton, linking dynamical image contraction to uniform chain stability and the continuity of $ω$-limit sets.

5. Extensions: Connectedness, Total Disconnectedness, and Boundary Cases

Moving beyond global shadowing, the relationship $X=LSC(ω_f)\iff X=CC(f)$ persists under more general settings, subject to connectedness and properties of $CR(f)$. Specifically, if $f:X\to X$ is continuous, $X$ connected, and $CR(f)$ totally disconnected, then the equivalence holds (Theorem 6.1). Here, the key insight is that, under these topological and dynamical constraints, the eventual image $S=\bigcap f^n(X)$ must be a singleton, and chain recurrence is forced into triviality, reflecting a rigid interplay between topological structure and dynamical regularity.

A plausible implication is that the mechanism rendering LSC and chain continuity equivalent is robust to the weakening of the shadowing hypothesis, provided that the chain recurrent set lacks internal topological complexity and the ambient space is connected.

6. Illustrative Examples and Counterexamples

To delineate the necessity of the various hypotheses in the main results, a collection of carefully constructed examples is provided:

Construction	Key Features	Theoretical Implication
Subshift of finite type	$X=C(ω_f)$, no shadowable points	Shadowing hypothesis is essential
$f(x)=x^2$ on $[0,1]$	$Sh(f)=X$, $USC(ω_f)=LSC(ω_f)=X\setminus\{1\}$	Sharpness of shadowable-point criteria
One-sided shift	Non-minimal $ω_f$ at some recurrent point	Shadowing is necessary for main equivalences
Skew rotation on $Y\times[0,1]$	$LSC(ω_f)=X$, $USC(ω_f)\subsetneq X$	USC/LSC can decouple; illustrates subtlety
Anosov automorphism on $\mathbb{T}^2$	$Sh(f)=\mathbb{T}^2$, $USC(ω_f)$ dense $G_δ$, $LSC(ω_f)=\emptyset$	Limitations on $LSC(ω_f)$
Irrational rotation on the circle	$X=C(ω_f)$ but $Sh(f)=\emptyset$	Shadowing can fail even for continuity
Denjoy counterexample on the circle	$CR(f)=X$ Cantor, $X=C(ω_f)$, $Sh(f)=\emptyset$, no LSC/chain continuity	Total disconnectedness insufficient alone

These examples clarify that shadowing, global shadowing, connectedness, and the structure of $CR(f)$ are not redundant but are each critical for the stated equivalences and continuity results to hold. Dropping any one can lead to failures of semicontinuity, chain stability, or minimality, revealing the sharpness of the established theorems (Kawaguchi, 13 Jan 2026).

7. Summary and Significance in Dynamical Systems

The continuity of $ω$-limit sets underlies the stability of asymptotic dynamical behavior. The explicit criteria relating upper and lower semicontinuity to shadowing and chain-continuity furnish a comprehensive framework for understanding the persistence of long-term dynamical features under perturbation and discrete approximations. The derived equivalences demonstrate that, on compact metric spaces, the behavior of $ω$-limit sets is tightly constrained by shadowing and the topology of the chain recurrent set. This foundational analysis informs the refined classification of dynamical systems and provides a template for further investigations into persistence phenomena and structural stability under varying topological and dynamical regimes (Kawaguchi, 13 Jan 2026).