Average Shadowing in Topological Dynamics

• Average Shadowing is a pseudo-orbit tracing property in topological dynamics that uses sliding-window Cesàro control to ensure every sufficiently accurate average pseudo-orbit is closely traced by a true orbit.
• It links classical shadowing with specification, chain dynamics, and invariant-measure theory, with several equivalent formulations in compact-metric settings.
• The concept underpins ergodic and mixing properties in systems such as shifts of finite type, Anosov homeomorphisms, and uniformly contracting IFS, influencing invariant measure realization and hyperbolicity.

Average shadowing is a pseudo-orbit tracing property in topological dynamics that replaces pointwise control of orbit-tracing errors by Cesàro control. In the standard compact-metric setting, it requires that every sufficiently accurate average pseudo-orbit be traced by a true orbit with small average discrepancy; this places it between classical shadowing and broader specification-like or density-based tracing regimes (Carvalho et al., 2014). The subject has developed into a substantial interface between shadowing theory, specification, chain dynamics, invariant-measure theory, and Besicovitch-type orbit metrics, with later work showing that several apparently distinct “average” shadowing notions are in fact equivalent in important settings (Kulczycki et al., 2013).

1. Formal definitions and basic framework

A common setting is a continuous map $f:X\to X$ on a compact metric space $(X,d)$. A one-sided sequence $(x_i)_{i\ge 0}$ is a $\delta$-pseudo-orbit if

$$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$

For average shadowing, the usual Blank formulation defines a $\delta$-average-pseudo-orbit by the sliding-window condition: there exists $N=N(\delta)$ such that for every $n\ge N$ and every $k\ge 0$,

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

The sequence $(X,d)$0 is $(X,d)$1-shadowed in average by $(X,d)$2 if

$(X,d)$3

The map $(X,d)$4 has the average shadowing property, usually abbreviated ASP, if for every $(X,d)$5 there exists $(X,d)$6 such that every $(X,d)$7-average-pseudo-orbit is $(X,d)$8-shadowed in average (Carvalho et al., 2014).

Closely related is the asymptotic average shadowing property (AASP). A sequence $(X,d)$9 is an asymptotic average pseudo-orbit if

$(x_i)_{i\ge 0}$0

and it is asymptotically shadowed in average by $(x_i)_{i\ge 0}$1 if

$(x_i)_{i\ge 0}$2

AASP requires that every asymptotic average pseudo-orbit admit such a tracer (Kulczycki et al., 2013).

Several papers also use limsup-based variants of average pseudo-orbits or definitions phrased directly in terms of $(x_i)_{i\ge 0}$3-pseudo-orbits rather than $(x_i)_{i\ge 0}$4-average-pseudo-orbits. These distinctions are not merely terminological: a substantial part of the theory is devoted to showing when such formulations coincide, when they differ, and which additional hypotheses force equivalence (Wu et al., 2014).

The natural metric framework for average tracing is the Besicovitch pseudometric. For sequences $(x_i)_{i\ge 0}$5 and $(x_i)_{i\ge 0}$6,

$(x_i)_{i\ge 0}$7

and for points $(x_i)_{i\ge 0}$8,

$(x_i)_{i\ge 0}$9

In this language, average shadowing is ordinary shadowing for the orbit map in a pseudometric that forgets sparse large deviations and retains only their asymptotic frequency and magnitude (Kwietniak et al., 2016).

2. Relation to specification, density shadowing, and other tracing notions

Average shadowing sits inside a now fairly precise hierarchy of specification-like and density-based properties. Earlier work on compact surjective systems proved

$\delta$0

and identified chain mixing as a key bridge between the asymptotic and non-asymptotic formulations (Kulczycki et al., 2013). A parallel development showed that almost specification also implies AASP without assuming surjectivity, and that AASP implies ASP and consequently $\delta$1-shadowing and $\delta$2-shadowing (Wu et al., 2014).

A useful summary of established implications is as follows.

Property relation	Scope	Source
almost specification $\delta$3 AASP $\delta$4 ASP $\delta$5 $\delta$6- and $\delta$7-shadowing	compact systems	(Wu et al., 2014)
weak specification $\delta$8 AASP	surjective systems	(Kwietniak et al., 2016)
vague specification $\delta$9 AASP	compact metrizable systems	(Can et al., 2024)
ASP, weak asymptotic average shadowing, mean ergodic shadowing, almost asymptotic average shadowing, AASP, and $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$0-shadowing are equivalent	continuous self-maps on compact metric spaces	(Das et al., 10 Apr 2025)
under shadowing, total transitivity, weak mixing, mixing, specification, almost specification, AASP, and ASP are equivalent	compact systems with shadowing	(Kulczycki et al., 2013)

Two later results substantially sharpened the landscape. First, vague specification was shown to be exactly AASP on compact metrizable spaces, making AASP one of the weakest specification-like properties currently isolated in the literature (Can et al., 2024). Second, for a continuous self-map on a compact metric space, ASP is not merely implied by AASP: it is equivalent to AASP and to several other average/density shadowing variants, answering a question posed in 2014 (Das et al., 10 Apr 2025).

Under the classical shadowing property, distinctions collapse even further. In that regime, total transitivity, weak mixing, topological mixing, surjective specification, surjective almost specification, surjective AASP, and surjective ASP are equivalent; in the $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$1-expansive case, periodic specification joins the same list (Kulczycki et al., 2013). This collapse explains why, in many expansive settings, average shadowing behaves less as an independent notion and more as one facet of a larger hyperbolic or specification-type package.

3. Dynamical and ergodic consequences

Average shadowing has strong topological consequences when combined with recurrence or shadowing hypotheses. In compact surjective systems, ASP implies chain mixing, and almost specification also implies chain mixing (Kulczycki et al., 2013). In the presence of the two-sided limit shadowing property for homeomorphisms, average shadowing arises together with shadowing, asymptotic average shadowing, specification, and topological mixing; the mechanism is structural, passing through shadowing and total transitivity rather than through explicit $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$2-estimates (Carvalho et al., 2014).

The ergodic consequences are equally strong. AASP implies that every invariant measure has a generic point. The construction proceeds by associating to each invariant measure $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$3 an asymptotic average pseudo-orbit $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$4 with $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$5; any point that asymptotically traces $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$6 in average is then generic for $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$7 because distribution measures are continuous with respect to $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$8 (Kwietniak et al., 2016).

ASP supports a broader realization theory for invariant measures. If $$d\big(f(x_i),x_{i+1}\big)<\delta \quad \text{for all } i\ge 0.$$9 is non-empty, compact, and connected, then there exists $\delta$0 such that

$\delta$1

where $\delta$2 is the set of subsequential weak$\delta$3 limits of empirical measures of $\delta$4. Moreover,

$\delta$5

is dense in

$\delta$6

and if $\delta$7 is isolated or equals $\delta$8, then

$\delta$9

is residual in $N=N(\delta)$0 (Dong et al., 2013). This is a Sigmund-type abundance theorem under ASP rather than full specification.

Measure-theoretic richness also appears through support conditions. If ASP holds and there exists a fully supported invariant measure, then the system is weakly mixing (Kulczycki et al., 2013). On the factor side, ASP forces the maximal equicontinuous factor to be a singleton, so compact distal systems with ASP are necessarily trivial (Kulczycki et al., 2013). These statements make precise a recurring theme: average shadowing is incompatible with substantial equicontinuous structure.

4. Canonical classes of systems and representative examples

In expansive symbolic and hyperbolic dynamics, ASP often appears as part of a larger equivalence class. For homeomorphisms with the two-sided limit shadowing property, topological mixing, specification, ASP, and AASP all follow. In particular, mixing shifts of finite type and topologically mixing Anosov homeomorphisms have the two-sided limit shadowing property and therefore satisfy ASP and AASP (Carvalho et al., 2014).

Symbolic examples also show that ASP is genuinely weaker than specification-type properties. A proximal mixing subshift was constructed with ASP, positive entropy, and a fully supported invariant measure, but without almost specification; more generally, there are systems with AASP but without weak or almost specification (Kwietniak et al., 2016). Minimality is likewise compatible with average shadowing: proximal and minimal shift spaces constructed later were shown to possess vague specification, hence AASP, and therefore average-shadowing-type behavior in the Besicovitch sense (Can et al., 2024).

At the generic level, on a compact connected manifold $N=N(\delta)$1 of dimension $N=N(\delta)$2, shadowing and periodic shadowing are generic both in the full $N=N(\delta)$3 space $N=N(\delta)$4 and in the conservative space $N=N(\delta)$5. In the conservative case, generic mixing is already known, and mixing plus shadowing yields specification, ASP, and AASP on a residual subset of $N=N(\delta)$6 (Guihéneuf et al., 2016). Thus, in $N=N(\delta)$7-generic conservative dynamics, average shadowing is not exceptional but typical.

Parameterized iterated function systems furnish another robust class. Uniformly contracting IFS have ASP, and the Sierpinski IFS is a concrete example. ASP is preserved under appropriate conjugacies and under products, and if a continuous surjective IFS on a compact metric space has ASP, then every point is chain recurrent and the IFS has a single chain component (Nia, 2015). These facts parallel single-map theory but do not collapse to it, since the circle example in the same paper shows that chain recurrence of every point does not imply ASP.

Noncompactness marks a genuine boundary. Earlier examples constructed metric systems $N=N(\delta)$8 and $N=N(\delta)$9 with maps $n\ge N$0 such that $n\ge N$1 has ASP but not AASP, while $n\ge N$2 has AASP but not ASP, showing that compactness is essential for the strongest implication chains (Kulczycki et al., 2013). Later compact equivalence theorems therefore resolve a specifically compact phenomenon rather than a completely general one.

5. Generalizations beyond autonomous single-map dynamics

Average shadowing extends coherently to several nonclassical dynamical frameworks. For finitely generated free semigroup actions directed by a fixed symbol sequence $n\ge N$3, the notions of $n\ge N$4-average shadowing, weak asymptotic average shadowing, mean ergodic shadowing, almost asymptotic average shadowing, $n\ge N$5-AASP, and $n\ge N$6-$n\ge N$7-shadowing for every $n\ge N$8 are equivalent on compact metric spaces (Das et al., 10 Apr 2025). This gives a semigroup-level analogue of the compact autonomous equivalence theorem.

For parameterized IFS, one can define asymptotic average pseudo-orbits relative to control sequences and ask for average tracing by possibly different control sequences. Every uniformly contracting IFS has AASP; AASP is invariant under topological conjugacy, stable under passage to powers and products, and for continuous surjective IFS on compact spaces it implies chain transitivity (Nia, 2015). Together with the ASP theory for IFS, this yields a substantial shadowing theory for multimap dynamics rather than just for single iterates.

Set-valued “multiple mappings” admit a Hausdorff-metric version of average shadowing. If $n\ge N$9 acts on $k\ge 0$0, then a sequence $k\ge 0$1 is a $k\ge 0$2-average-pseudo-orbit when

$k\ge 0$3

for all sufficiently large $k\ge 0$4 and all $k\ge 0$5, and ASP requires average tracing by an orbit $k\ge 0$6 in the Hausdorff metric. In this setting, ASP passes from $k\ge 0$7 to all iterates $k\ge 0$8 and implies chain transitivity, but it is not inherited merely because each component map has ASP (Zhao, 2023).

Nonautonomous systems also support average-shadowing-type inheritance results. For a sequence of onto maps $k\ge 0$9 on compact metric spaces, asymptotic average shadowing on a closed invariant subset $$\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_{i+k}),x_{i+k+1}\big)<\delta.$$0 lifts to the whole system provided a density condition ensures that every orbit spends asymptotically almost all time near $$\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_{i+k}),x_{i+k+1}\big)<\delta.$$1 (An, 2024). This is the nonautonomous counterpart of measure-center lifting phenomena in autonomous systems.

6. Robustness, obstructions, and current frontiers

In conservative smooth dynamics, robust average shadowing has strong geometric consequences. For $$\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_{i+k}),x_{i+k+1}\big)<\delta.$$2-robust incompressible flows, average shadowing or asymptotic average shadowing implies dominated splitting on the whole manifold, while robust limit shadowing implies that the flow is transitive Anosov. In the Hamiltonian case, $$\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_{i+k}),x_{i+k+1}\big)<\delta.$$3-stable average or asymptotic average shadowing implies partial hyperbolicity on the regular energy surface, and stable limit shadowing implies Anosov behavior (Bessa et al., 2013). Thus, in conservative flow settings, robust average shadowing is a hyperbolicity-forcing property.

There are also sharp obstructions. Geometric Lorenz flows have neither average shadowing, nor asymptotic average shadowing, nor limit shadowing when reparametrizations are taken from the class of increasing homeomorphisms with the bounded-deviation condition used in that work (Arbieto et al., 2013). More generally, for isolated invariant sets of $$\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_{i+k}),x_{i+k+1}\big)<\delta.$$4 flows, average shadowing and asymptotic average shadowing exclude proper attractors, and two-sided limit shadowing with a gap implies topological transitivity and the classical shadowing property (Chu et al., 2016).

Recent work has tightened the connection with specification. Partial specification implies ASP on compact dynamical systems; if the system is surjective and has partial specification, then ergodic invariant measures are dense in the space of invariant measures (Can et al., 14 Apr 2026). This fits the general pattern that average shadowing often emerges from weakened gluing properties long before full Bowen specification becomes available.

Historically, one open question asked whether shadowing plus specification implies the two-sided limit shadowing property. That question remains explicitly recorded in the literature in connection with the strength of two-sided limit tracing beyond expansive settings (Carvalho et al., 2014). By contrast, the older question whether ASP implies AASP has been answered affirmatively for continuous self-maps on compact metric spaces and for finitely generated free semigroup actions (Das et al., 10 Apr 2025). The remaining frontier is therefore less about the internal equivalence of average-shadowing variants on compact spaces and more about how far these equivalences extend under weaker compactness, weaker regularity, or more general gluing mechanisms.