Limit Shadowing in Dynamical Systems

• Limit shadowing is a property in topological dynamical systems that ensures asymptotic traceability of pseudo-orbits by true orbits of continuous maps.
• It generalizes classical shadowing by allowing errors to vanish over time and includes variants such as s-limit and two-sided limit shadowing with distinct rigidity effects.
• This property underpins the classification of chaotic dynamics, chain components, and contributes to understanding structural stability in both autonomous and nonautonomous settings.

Limit shadowing, in the context of topological dynamical systems, is a property describing the asymptotic traceability of approximate orbits—so-called “pseudo-orbits”—by true orbits of a continuous map. Explicitly, a continuous map $f$ on a compact metric space $(X, d)$ is said to have the limit shadowing property if for every sequence $(x_i)_{i \geq 0}$ satisfying $\lim_{i \to \infty} d(f(x_i), x_{i+1}) = 0$, there exists $y \in X$ such that $\lim_{i \to \infty} d(f^i(y), x_i) = 0$. This notion generalizes the classical shadowing property by relaxing uniform error control to an asymptotic one and is central to understanding the robustness of recurrent dynamics under vanishing perturbations. Several variants, such as $s$-limit shadowing and two-sided limit shadowing, further refine or strengthen the concept in discrete and continuous dynamical settings.

1. Formal Definitions and Variants

Let $(X, d)$ be a compact metric space and $f : X \to X$ continuous.

• Limit shadowing property: $f$ has the limit shadowing property if every limit-pseudo-orbit—that is, every sequence $(x_i)$ with $\lim_{i \to \infty} d(f(x_i), x_{i+1}) = 0$—is asymptotically shadowed by a true orbit; i.e., there exists $y \in X$ such that $\lim_{i \to \infty} d(f^i(y), x_i) = 0$ (Kawaguchi, 2017, Good et al., 2017, Good et al., 2019).
• $s$-limit shadowing: A map $f$ has $s$-limit shadowing if for every $\varepsilon > 0$ there exists $\delta > 0$ such that every $\delta$-limit-pseudo-orbit (i.e., $d(f(x_i),x_{i+1}) \leq \delta$ for all $i \geq 0$ and $\lim_{i \to \infty} d(f(x_i),x_{i+1}) = 0$) is $\varepsilon$-limit-shadowed by some $y \in X$, that is, $d(f^i(y), x_i) \leq \varepsilon$ for all $i$ and $\lim_{i \to \infty} d(f^i(y), x_i) = 0$ (Kawaguchi, 2023, Kawaguchi, 2024).
• Two-sided limit shadowing: For homeomorphisms or flows, the two-sided version requires that all bi-infinite limit pseudo-orbits $(x_i)_{i \in \mathbb{Z}}$ (with $\lim_{|i| \to \infty} d(f(x_i), x_{i+1}) = 0$) be shadowed in both forward and backward time (Carvalho et al., 2014, Carvalho, 2013, Aponte et al., 2019).

These variants strictly refine the classical (uniform) shadowing property. In particular, $s$-limit shadowing $\implies$ limit shadowing and classical (Bowen) shadowing, but not conversely (Kawaguchi, 2024, Kawaguchi, 2023, Good et al., 2017).

2. Relation to Shadowing, Chain Components, and Conley Decomposition

Limit shadowing sits within a hierarchy of pseudo-orbit tracing properties. Classical shadowing requires every approximate trajectory with uniformly small errors to be uniformly shadowed; limit shadowing permits errors only to decay asymptotically. $s$-limit shadowing synthesizes both requirements: it demands uniform smallness and asymptotic vanishing of pseudo-orbit errors, enforcing strong rigidity.

The existence of the limit shadowing property has significant structural consequences for the dynamics. For instance, if $f$ has the limit shadowing property, then the restriction $f|_{\Omega(f)}$ to the non-wandering set has the shadowing property (Kawaguchi, 2017). In equicontinuous systems, limit shadowing and shadowing are equivalent and also imply that the non-wandering set is totally disconnected.

The $s$-limit shadowing property further enforces that every chain component admits classical shadowing; this is formalized in the result $\mathcal{C}(f) = \mathcal{C}_{sp}(f)$: all chain components support the shadowing property, where $\mathcal{C}_{sp}(f)$ denotes the set of chain components on which $f$ restricts to a map with shadowing (Kawaguchi, 2023). Such chain-recurrent decompositions, especially via $G_\delta$ partitions (into basins $W(C)$ and stable pieces $V(D)$), allow for precise analysis of complex behaviors in each dynamical region (Kawaguchi, 2024).

3. Global $G_\delta$ Partition and Dynamical Implications

A continuous self-map $f$ with $s$-limit shadowing on a compact metric space admits a refinement of the Conley decomposition of $X$:

$$X = \bigsqcup_{C \in \mathcal{C}(f)} \bigsqcup_{D \in D(C)} V(D),$$

where:

• $\mathcal{C}(f)$ are the chain components (closed, $f$-invariant, chain transitive),
• $D(C)$ are equivalence classes under a chain-proximal relation in each $C$,
• $V(D)$ are $G_\delta$-sets defined as those $x \in W(C)$ for which $\lim_{i \to \infty} d(f^i(x), f^i(D)) = 0$ (Kawaguchi, 2024).

This partition is dynamically meaningful: within each $V(D)$, the $s$-limit shadowing property ensures the existence of points whose forward trajectories are asymptotically close, allowing robust approximation and shadowing. Lemma 1.1 of (Kawaguchi, 2024) quantifies this by stating that for every $x, y \in V(D)$ and every $\varepsilon > 0$, there exists $z \in V(D)$ such that $d(y, z) \leq \varepsilon$ and $\lim_{i \to \infty} d(f^i(x), f^i(z)) = 0$.

4. Impact on Chaos and Scrambled Sets

Limit shadowing, particularly in its $s$-limit version, has profound implications on the existence and genericity of various types of chaos, especially in the formulation via Furstenberg families and Li–Yorke–type scrambled sets (Kawaguchi, 2024, Kawaguchi, 2023). For $n \geq 2$ and $r > 0$, define

$$S_f(x_1, ..., x_n; r) = \left\{\, i \in \mathbb{N}: \min_{1 \leq j < k \leq n} d(f^i(x_j), f^i(x_k)) > r \,\right\},$$

$$T_f(x_1, ..., x_n; \epsilon) = \left\{\, i \in \mathbb{N}: \max_{1 \leq j < k \leq n} d(f^i(x_j), f^i(x_k)) < \epsilon \,\right\}.$$

An $n$-tuple is $(\mathscr{F}, \mathscr{G})$-$\delta$-scrambled if $S_f$ belongs to the Furstenberg family $\mathscr{F}$, and $T_f$ to $\mathscr{G}$ for all small $\epsilon > 0$.

Under $s$-limit shadowing, the existence of an $n$-tuple in some $V(D)$ with $S_f \in \mathscr{F}$ leads to the density or residuality (in the sense of Baire category) of $(\mathscr{F}, \mathscr{G})$-scrambled $n$-tuples in every $V(E)$ for $E \in D(C)$ (Kawaguchi, 2024). Consequently, $s$-limit shadowing guarantees the robust presence of Li–Yorke–type chaos, with multi-scrambled uncountable sets often constructed via Mycielski's theorem.

5. Conjugacy, Inheritance, and Nonautonomous/Time-Varying Contexts

The limit shadowing property, as well as its $s$-limit and $h$-shadowing refinements, is invariant under (uniform) topological conjugacy (Sarkooh, 2021). For time-varying maps and nonautonomous dynamical systems, suitable generalizations admit analogous definitions and transfer arguments. In particular, for sequences of onto continuous maps $(f_n)$ subject to strong equicontinuity and classical shadowing, limit shadowing persists (An, 2024). Furthermore, for strongly expansive systems with classical shadowing, all stronger forms (including $h$-, $s$-limit, and limit shadowing) hold (Sarkooh, 2021). Limit shadowing is preserved under standard inverse limits, Cartesian products, and under certain factor maps with an almost-lifting property for asymptotic pseudo-orbits (Good et al., 2019).

6. Illustrative Examples, Counterexamples, and Classification Results

The spectrum of systems with (or without) limit shadowing is delineated through canonical examples:

• Piecewise linear interval maps with constant slope: For tent maps and other maps with the so-called linking property, shadowing, $s$-limit shadowing, and limit shadowing all coincide (Good et al., 2017).
• Cantor set homeomorphisms: The identity map on a Cantor set trivially has $s$-limit shadowing, and all chain components are singletons supporting shadowing (Kawaguchi, 2023).
• Systems with SP but not limit shadowing: Constructed via fibered subshifts or specific interval restrictions, demonstrating separation of these properties (Good et al., 2017).
• Anosov diffeomorphisms: Two-sided limit shadowing characterizes transitive Anosov systems; in particular, the $C^1$-interior of the set of diffeomorphisms with the two-sided limit shadowing property is precisely the set of transitive Anosov diffeomorphisms (Carvalho, 2013, Carvalho, 2015).
• Genericity phenomena: $s$-limit shadowing is $C^0$-dense (but not generic) in the space $C(M)$ of continuous self-maps of a manifold, but is not generic among homeomorphisms on a closed differentiable manifold (Kawaguchi, 2023).

7. Concluding Synthesis and Open Problems

Limit shadowing is a cornerstone property in the theory of dynamical stability, offering a bridge between uniform pseudo-orbit tracing and robust recurrence. The $s$-limit shadowing property, sitting strictly above limit shadowing and classical shadowing, guarantees a highly organized, dynamically rich decomposition of the phase space, and underpins the prevalence of structured chaos, notably in the presence of generic families of Furstenberg-type scrambled sets (Kawaguchi, 2024).

Outstanding questions concern the characterization of systems for which weaker shadowing notions suffice to produce similarly global chaotic decompositions, the full classification of chain components supporting multi-scrambled sets, and the delineation of settings (particularly in the nonautonomous and noncompact regime) where limit shadowing is stable under natural dynamical operations. The relationship between two-sided limit shadowing and specification, as well as detailed classifications in the category of $C^1$-generic diffeomorphisms, remain prominent areas of ongoing research.