Average Shadowing in Dynamical Systems

• Average shadowing is a property in dynamical systems that ensures pseudo-orbits with small averaged errors are globally approximated by true trajectories.
• It relaxes the strict, pointwise error control of classical shadowing by focusing on time-averaged measures, applying to systems like discrete maps, flows, and iterated function systems.
• The concept connects with ergodic theory, specification, and robustness, offering both theoretical insights and practical numerical methods for sensitivity analysis.

Average shadowing is a property in topological and metric dynamics that captures the ability of a system to globally shadow approximate orbits—called pseudo-orbits—when the errors are small in an averaged, rather than pointwise, sense. Unlike the classical shadowing property, which requires the system to shadow any sufficiently small local error sequence uniformly, average shadowing relaxes this to constraining only the time-average of the errors. Average shadowing, along with its several variants (asymptotic, almost, weak, mean ergodic, $M_\alpha$, etc.), has been extensively studied across discrete maps, flows, non-autonomous systems, free semigroup actions, iterated function systems (IFS), and set-valued dynamics. This property has robust connections to specification, generic points, mixing, chain recurrence, and ergodic theory.

1. Formal Definitions and Main Variants

Let $(X,d)$ be a compact metric space and $f:X\to X$ a continuous map. Given $\delta>0$, a sequence $(x_n)_{n=0}^\infty$ is a $\delta$-average pseudo-orbit if there exists $N\in\mathbb N$ such that for all $n \geq N$ and all $k \geq 0$,

$$\frac{1}{n}\sum_{i=0}^{n-1} d\left(f(x_{i+k}), x_{i+k+1}\right) < \delta.$$

Given $\epsilon>0$, $(x_n)$ is $\epsilon$-shadowed in average by $y\in X$ if

$$\limsup_{n\to\infty} \frac{1}{n} \sum_{i=0}^{n-1} d(f^i(y),x_i) < \epsilon.$$

The map $f$ has the average shadowing property (ASP) if for every $\epsilon > 0$ there exists $\delta > 0$ such that every $\delta$-average pseudo-orbit is $\epsilon$-shadowed in average by some $y\in X$.

The asymptotic average pseudo-orbit notion relaxes the requirement to: $$\lim_{n\to\infty} \frac{1}{n} \sum_{i=0}^{n-1} d(f(x_i), x_{i+1}) = 0.$$ A system has the asymptotic average shadowing property (AASP) if every asymptotic average pseudo-orbit is asymptotically shadowed in average by some orbit. Further variants include the weak asymptotic average shadowing property, almost average shadowing property, and mean ergodic shadowing, each controlling the density or statistical distribution of error indices in different ways (Kulczycki et al., 2013, Wu et al., 2014, Kwietniak et al., 2016, Das et al., 10 Apr 2025, Garg et al., 2016).

For continuous-time systems (flows) and non-autonomous or set-valued dynamics, the definitions are adapted to cover time-varying maps, Hausdorff distances, or parameter sequences (An, 2024, Zhao, 2023, Bessa et al., 2013, Nia, 2015).

2. Hierarchies, Equivalences, and Relationships to Specification

In compact, surjective dynamical systems, there exists a strict implication chain: $$\text{almost specification} \implies \text{AASP} \implies \text{ASP} \implies \text{chain mixing} \implies \text{transitivity}$$ (Kulczycki et al., 2013, Wu et al., 2014). Classical specification is stronger than almost specification and thus implies all average shadowing properties.

For finitely generated free semigroup actions, the six principal average-type shadowing properties (average, weak asymptotic, mean ergodic, almost asymptotic, $M_\alpha$, asymptotic average shadowing) are equivalent (Das et al., 10 Apr 2025). Analogous equivalences have been characterized for symbolic dynamics and other expansive homeomorphisms (Carvalho et al., 2014).

On compact metrizable spaces, AASP always implies ASP, but the converse is generally false unless additional completeness (e.g., with respect to the dynamical Besicovitch pseudometric) is assumed (Can et al., 2024). In surjective, Besicovitch-complete systems, ASP and AASP do coincide.

If a system admits weak or vague specification, this always implies AASP (and thus ASP as well). Vague specification is shown to be equivalent to AASP and strictly weaker than weak specification (Kwietniak et al., 2016, Can et al., 2024).

3. Dynamical and Ergodic Consequences

Average and asymptotic average shadowing enforce strong global dynamical features, especially on compact spaces:

• Chain Transitivity: ASP or AASP implies chain transitivity (and hence chain mixing if total average chain transitivity holds) (Garg et al., 2016, Nia, 2015, Zhao, 2023).
• Ergodic Theoretic Structure: AASP entails the existence of generic points for every invariant measure; more strongly, the set of generic points for an ergodic measure is $D_B$-closed, where $D_B$ is the Besicovitch pseudometric (Kwietniak et al., 2016).
• Mixing and Uniqueness of Chain Components: Under compactness, AASP/ASP yields that the chain recurrent set forms a single chain component, and mixing follows if the space is also surjective (Garg et al., 2016, Kulczycki et al., 2013).
• Prohibitions on Attractors: In continuous flows, the AASP or ASP on an isolated invariant set implies the absence of proper attractors, implying chain transitivity (Chu et al., 2016).

Table: Implications for Compact, Surjective $f: X \to X$ | Property | Implies | Reference | |------------------------------|---------------------|--------------------| | Specification | ASP, AASP | (Kulczycki et al., 2013) | | AASP | ASP, Chain mixing | (Kulczycki et al., 2013, Garg et al., 2016) | | ASP | Chain mixing | (Kulczycki et al., 2013) |

Under two-sided limit shadowing or transitive specification, all shadowing-type and specification-type properties coincide (Carvalho et al., 2014).

4. Generalizations: Non-Autonomous, Set-Valued, and Semigroup Actions

Average shadowing has been extended to diverse settings:

• Non-Autonomous Dynamics: For sequences of expanding maps with vanishing product compression ratio, one obtains unique shadowing of pseudo-orbits and extension of asymptotic average shadowing from invariant subsystems under explicit density-recurrence conditions (An, 2024).
• Set-Valued and IFS Dynamics: The definition is adapted to compact set sequences under the Hausdorff metric; average shadowing in this setting implies chain transitivity but is strictly stronger than chain recurrence (Nia, 2015, Zhao, 2023).
• Parametrized IFS and Free Semigroup Actions: Through appropriate control sequences or semigroup words, average shadowing is robust under conjugacy, products, and iteration (Nia, 2015, Das et al., 10 Apr 2025).
• Flows: For flows, the Cesàro-averaged version of shadowing is defined through time-averaged integrals along reparameterized orbits; results for robust average or asymptotic average shadowing guarantee partial hyperbolicity or dominated splitting (Anosov in the limit shadowing case) in conservative or Hamiltonian flows (Bessa et al., 2013).

Notably, systems with singularities—such as the geometric Lorenz flow—may lack all standard forms of average shadowing; the lack of average or limit shadowing is intimately connected to failure of uniform hyperbolicity (Arbieto et al., 2013).

5. Structural and Generic Properties

Average shadowing is generic for homeomorphisms of compact manifolds (with or without measure-preservation), as established via Markovian intersection and chaining arguments. In particular, generic conservative homeomorphisms enjoy the shadowing, periodic shadowing, specification, ASP, and AASP properties (Guihéneuf et al., 2016). The presence of an average shadowing property is also robust under products and passage to subsystem closures, provided certain invariance or recurrence conditions hold (Kulczycki et al., 2013). On the measure center, almost specification lifts average shadowing and vice versa (Wu et al., 2014).

A constructive mechanism called the "gluing property" unifies and explains many shadowing results: if a single-gap gluing rate with summable tail exists, then the system exhibits average (and possibly uniform) shadowing even for discontinuous or non-invertible maps (Blank, 2022, Blank, 2022).

6. Connections to Other Dynamical Notions and Open Problems

Average shadowing interacts non-trivially with other orbit-tracing and recurrence notions:

• Obstructions: It is incompatible with nontrivial equicontinuous minimal systems—no nontrivial equicontinuous map has average shadowing (Wu et al., 2014).
• Non-Compact Counterexamples: The equivalence of AASP and ASP, and implications to chain mixing, may fail without compactness (Kulczycki et al., 2013).
• Invariant Set Extensions and Factors: ASP and AASP pass from invariant measure centers and are inherited by factors under vague specification, but these transfers are not always reversible for arbitrary subsystems (Wu et al., 2014, Can et al., 2024).

Important open problems include the full equivalence of ASP and AASP beyond Besicovitch-complete spaces, inheritance of ASP/AASP under factor maps in general, and the existence of nontrivial minimal systems exhibiting these properties (Can et al., 2024, Kulczycki et al., 2013).

7. Computational and Applied Aspects

Average shadowing provides a foundation for the rigorous analysis of numerical orbit approximations, especially in the study of chaotic systems. In hyperbolic and certain non-hyperbolic systems (e.g., via the periodic shadowing and least-squares shadowing algorithms), accurate bounded shadowing directions can be computed for sensitivity analysis of time-averaged observables, with error rates characterized by $O(1/T)$ and $O(1/\sqrt{T})$ as the averaging window increases (Lasagna et al., 2018). However, structural instabilities and non-uniform hyperbolicity (as in the Lorenz system) manifest as non-negligible shadowing errors or biases, emphasizing the dynamical relevance of strong shadowing properties.

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References:

• (Kulczycki et al., 2013) On almost specification and average shadowing properties
• (Wu et al., 2014) On various definitions of shadowing with average error in tracing
• (Kwietniak et al., 2016) Generic Points for Dynamical Systems with Average Shadowing
• (Can et al., 2024) On the weakness of the vague specification property
• (Das et al., 10 Apr 2025) Equivalence of Variants of Shadowing of Free Semigroup Actions
• (Garg et al., 2016) Average chain transitivity and the almost average shadowing property
• (An, 2024) The Shadowing Properties Of Nonautonomous Dynamical System
• (Blank, 2022) Average shadowing and gluing property
• (Blank, 2022) Average shadowing revisited
• (Zhao, 2023) Shadowing, average shadowing and transitive properties of multiple mappings
• (Nia, 2015) Iterated function systems with the average shadowing property
• (Nia, 2015) Parameterized IFS with the asymptotic average shadowing property
• (Bessa et al., 2013) Conservative flows with various types of shadowing
• (Guihéneuf et al., 2016) On the genericity of the shadowing property for conservative homeomorphisms
• (Lasagna et al., 2018) Periodic Shadowing Sensitivity Analysis of Chaotic Systems
• (Arbieto et al., 2013) On Various Types of Shadowing for Geometric Lorenz Flows
• (Chu et al., 2016) A note on shadowing properties
• (Carvalho et al., 2014) On homeomorphisms with the two-sided limit shadowing property