Average Shadowing Property (ASP)

• Average Shadowing Property (ASP) is a dynamical condition requiring that every δ-average pseudo-orbit is ε-shadowed in a mean (Cesàro) sense by a true orbit.
• ASP extends classical shadowing by using average (Cesàro) convergence and is equivalent to formulations like asymptotic average shadowing and Besicovitch pseudometric criteria.
• Its applications span invariant measure construction, specification properties, and analysis of systems such as nonautonomous maps, iterated function systems, and conservative flows.

Average Shadowing Property (ASP) is a shadowing-type condition in which pseudo-orbits are required to be traced only in Cesàro average rather than pointwise. In the standard compact-map setting, a system $(X,T)$ has ASP if for every $\varepsilon>0$ there exists $\delta>0$ such that every $\delta$-average pseudo-orbit is $\varepsilon$-shadowed in average by some point. Closely related notions include asymptotic average shadowing, weak asymptotic average shadowing, density-based shadowing, and Besicovitch-pseudometric formulations. The subject now spans compact dynamical systems, conservative flows, discontinuous and non-invertible maps, iterated function systems, free semigroup actions, and set-valued multiple mappings, with strong links to specification-like properties, generic-point theory, and chain-transitivity phenomena (Kwietniak et al., 2016, Wu et al., 2014, Bessa et al., 2013, Das et al., 10 Apr 2025).

1. Foundational definitions and formulations

For a compact dynamical system $(X,T)$, with $X$ a compact metric space with metric $\rho$, a sequence $\{x_n\}_{n=0}^\infty$ is a $\delta$-average pseudo-orbit if there exists $\varepsilon>0$0 such that for every $\varepsilon>0$1 and every $\varepsilon>0$2,

$\varepsilon>0$3

A point $\varepsilon>0$4 $\varepsilon>0$5-shadows $\varepsilon>0$6 in average if

$\varepsilon>0$7

ASP then requires that every sufficiently accurate average pseudo-orbit be shadowed in this sense (Kwietniak et al., 2016).

The asymptotic version replaces fixed small average error by vanishing average error. A sequence is an asymptotic average pseudo-orbit if

$\varepsilon>0$8

and it is asymptotically shadowed in average by $\varepsilon>0$9 if

$\delta>0$0

The corresponding system-level property is the asymptotic average shadowing property (AASP), which is frequently the stronger hypothesis used in structural theorems (Kwietniak et al., 2016).

A central geometric reformulation uses the Besicovitch pseudometric

$\delta>0$1

With this notation, asymptotic average pseudo-orbits are exactly the sequences $\delta>0$2 satisfying

$\delta>0$3

and asymptotic tracing in average by a point $\delta>0$4 is exactly

$\delta>0$5

where $\delta>0$6 (Kwietniak et al., 2016).

Later work broadened the setting from compact continuous maps to arbitrary discrete-time systems $\delta>0$7 on metric spaces, explicitly allowing non-invertible and discontinuous maps. In that framework, average shadowing is expressed by

$\delta>0$8

for bi-infinite trajectories, or by the one-sided Cesàro analogue when the backward part is finite (Blank, 2022, Blank, 2022). This extension shifted ASP from a primarily hyperbolic-topological notion to a more flexible orbit-tracing principle.

2. Equivalences and the surrounding hierarchy

A major development in the theory is that ASP admits several equivalent formulations. Kulczycki, Kwietniak, and Oprocha proved that for a compact dynamical system $\delta>0$9, ASP is equivalent to the weak asymptotic average shadowing property and to having the $\delta$0-shadowing property for every $\delta$1; they also showed that ASP is equivalent to a formulation in which every sufficiently accurate $\delta$2-asymptotic-average-pseudo-orbit is $\delta$3-shadowed in average (Wu et al., 2014). In this language, ASP can be read either as a mean-distance condition or as a density-of-good-times condition.

The same circle of ideas was pushed much further for finitely generated free semigroup actions. For a fixed generator word $\delta$4, the average shadowing property, weak asymptotic average shadowing, mean ergodic shadowing, almost asymptotic average shadowing, $\delta$5-shadowing for every $\delta$6, and asymptotic average shadowing are all equivalent (Das et al., 10 Apr 2025). In particular, the longstanding question whether average shadowing implies asymptotic average shadowing receives an affirmative answer in that compact free-semigroup setting (Das et al., 10 Apr 2025).

The implication structure connecting ASP to specification-like properties is nuanced. Almost specification implies AASP, and AASP implies ASP, in compact systems; under ordinary shadowing, ASP, AASP, specification, almost specification, total transitivity, weak mixing, and topological mixing become equivalent after the appropriate surjectivity hypotheses are imposed (Kulczycki et al., 2013). Weak specification implies AASP, but AASP does not imply weak specification, nor almost specification (Kwietniak et al., 2016). More recently, partial specification was shown to imply ASP (Can et al., 14 Apr 2026).

A further simplification comes from vague specification. Vague pseudo-orbits are defined by density-one recurrence of shifted tails into every neighborhood of the orbit space $\delta$7, and vague specification requires that every such sequence be traced by a true orbit in Besicovitch pseudometric. The paper on the weakness of vague specification proves that vague specification is equivalent to AASP, and for surjective systems it proves

$\delta$8

This places ASP precisely at the interface between specification-like tracing and metric completeness in dynamical Besicovitch geometry (Can et al., 2024).

3. Measure-theoretic and ergodic consequences

ASP and especially AASP have strong consequences for invariant measures. A foundational result is that for every invariant measure $\delta$9 of a compact dynamical system one can construct an asymptotic average pseudo-orbit whose distribution measures equal $\varepsilon$0; any point asymptotically tracing that pseudo-orbit in average is then generic for $\varepsilon$1. Consequently, every invariant measure has a generic point in any system with AASP (Kwietniak et al., 2016).

The mechanism behind this conclusion is Besicovitch continuity of empirical-measure accumulation sets. If two sequences are close in $\varepsilon$2, then their sets of distribution measures are close in Hausdorff metric, and in particular $\varepsilon$3 implies

$\varepsilon$4

This makes asymptotic average tracing a measure-preserving operation at the level of distribution measures (Kwietniak et al., 2016).

ASP also supports much richer statistical realization phenomena. The abstract of the 2013 paper on ergodic properties states that for a system with ASP, every non-empty, compact and connected subset $\varepsilon$5 coincides with $\varepsilon$6 for some point $\varepsilon$7, where $\varepsilon$8 is the accumulation set of empirical measures of the orbit of $\varepsilon$9. The same abstract states that the set

$(X,T)$0

is dense in

$(X,T)$1

and that under an additional isolation hypothesis or when $(X,T)$2, the set of points whose empirical measures accumulate on all invariant measures is residual in $(X,T)$3 (Dong et al., 2013). This places ASP in the Sigmund-style tradition of realizing prescribed measure-theoretic behavior by single orbits.

A related consequence appears in the 2026 paper on partial specification and Besicovitch completeness: if $(X,T)$4 is surjective and has ASP, then ergodic measures are dense among invariant measures (Can et al., 14 Apr 2026). In conjunction with partial specification $(X,T)$5 ASP, this yields a specification-free route to density of ergodic measures.

4. Mechanisms proving average shadowing

One major proof paradigm is orbit gluing. In the general metric-space framework of “Average shadowing and gluing property,” the decisive hypothesis is a gluing property with rate $(X,T)$6, where a single true trajectory bridges the backward part of one orbit and the forward part of another, and the rate is summable: $(X,T)$7 Under that hypothesis, the paper proves standard shadowing $(X,T)$8 and, in the bounded-perturbation regime, average shadowing of rare perturbations $(X,T)$9 (Blank, 2022). The same paper explicitly identifies gluing as the abstraction of “shadowing for a single perturbation.”

The later paper “Average shadowing revisited” sharpens this approach. For arbitrary maps on metric spaces, including discontinuous and non-invertible ones, a summable strong gluing property implies both standard shadowing and weak average shadowing $X$0, where the pseudo-orbit errors are controlled only by

$X$1

The proof uses a parallel gluing scheme, dyadic block organization, and product estimates controlling the growth of intermediate gaps (Blank, 2022). This framework clarifies that ASP can arise from summable local orbit-bridging rather than from global hyperbolicity.

A different mechanism comes from Besicovitch geometry. In the compact-setting paper on generic points, average shadowing is naturally expressed in the pseudometric $X$2, and the map

$X$3

from sequences to distribution measures is uniformly continuous with respect to $X$4 (Kwietniak et al., 2016). This makes the shadowing problem compatible with empirical-measure convergence, which explains why AASP is so effective in generic-point and invariant-measure constructions.

The specification family provides a third mechanism. Almost specification implies AASP, weak specification implies AASP, vague specification is equivalent to AASP, and partial specification implies ASP (Kulczycki et al., 2013, Kwietniak et al., 2016, Can et al., 2024, Can et al., 14 Apr 2026). Taken together, these results show that ASP is not tied to a single combinatorial definition; rather, it is the shadowing-theoretic manifestation of several orbit-concatenation schemes of varying strength.

5. Extensions beyond a single continuous self-map

ASP has been extended to several nonclassical settings. For multiple mappings $X$5 on a compact metric space, the theory is formulated on the hyperspace $X$6 of nonempty compact subsets, using the Hausdorff metric $X$7. A sequence $X$8 is a $X$9-average-pseudo-orbit if long tail averages of $\rho$0 are $\rho$1, and ASP means that some point $\rho$2 satisfies

$\rho$3

In this set-valued context, ASP is preserved by iterative action $\rho$4 and implies chain transitivity, but it need not be inherited from the individual generators: there are continuous maps $\rho$5 each having shadowing and ASP while the multiple mapping $\rho$6 has neither (Zhao, 2023).

For parameterized iterated function systems, ASP and AASP both admit natural branch-dependent formulations. Uniformly contracting IFS have AASP (Nia, 2015) and uniformly contracting IFS also have ASP (Nia, 2015). In the ASP paper for IFS, continuous surjective IFS on compact metric spaces satisfy a strong chain consequence: if the IFS has ASP, then every point is chain recurrent and there is only one chain component (Nia, 2015). In the AASP paper, continuous surjective IFS with AASP are chain transitive (Nia, 2015). The Sierpinski IFS is explicitly identified as an IFS with ASP (Nia, 2015).

For nonautonomous systems, the 2024 paper develops AASP rather than standard ASP. It proves that if $\rho$7 has asymptotic average shadowing on an invariant closed subset $\rho$8, then $\rho$9 has asymptotic average shadowing under a density condition requiring that every orbit spend proportion $\{x_n\}_{n=0}^\infty$0 of a common time window near $\{x_n\}_{n=0}^\infty$1 (An, 2024). This suggests that the average-shadowing framework can be localized and then lifted in nonautonomous settings.

Conservative flows form another extension. For incompressible flows, $\{x_n\}_{n=0}^\infty$2-stable average shadowing or $\{x_n\}_{n=0}^\infty$3-stable asymptotic average shadowing implies a dominated splitting on the whole manifold; for Hamiltonian systems, the corresponding robust properties imply partial hyperbolicity on the regular energy surface (Bessa et al., 2013). Robust limit shadowing is even stronger, forcing Anosov behavior in both incompressible and Hamiltonian categories (Bessa et al., 2013). Here ASP functions not merely as an approximation property but as a rigidity indicator.

6. Structural consequences, genericity, and limitations

ASP interacts strongly with transitivity and mixing, but the implications depend on auxiliary hypotheses. In compact systems with ordinary shadowing, Kulczycki, Kwietniak, and Oprocha proved that total transitivity, topological weak mixing, topological mixing, surjective specification, surjective almost specification, surjective AASP, and surjective ASP are equivalent (Kulczycki et al., 2013). In the presence of shadowing, ASP therefore sits on the same structural level as the classical strong recurrence properties.

Two-sided limit shadowing is stronger still. For homeomorphisms of compact metric spaces, two-sided limit shadowing implies shadowing, average shadowing, asymptotic average shadowing, and specification (Carvalho et al., 2014). The route is indirect: two-sided limit shadowing yields shadowing and total transitivity, and a theorem quoted from Kulczycki–Kwietniak–Oprocha then identifies total transitivity with ASP, AASP, mixing, and specification under shadowing (Carvalho et al., 2014).

Genericity results also exist. The paper on conservative homeomorphisms proves generic shadowing and periodic shadowing in the conservative category, then combines this with generic topological mixing and known implication theorems to conclude that a generic conservative homeomorphism has the specification property, the average shadowing property, and the asymptotic average shadowing property (Guihéneuf et al., 2016). Thus ASP is not exceptional in conservative $\{x_n\}_{n=0}^\infty$4 dynamics.

At the same time, ASP is not a synonym for topological chaos without qualification. Compact examples with almost specification, AASP, and ASP can still be nontransitive or proximal (Kulczycki et al., 2013). For multiple mappings, ASP of each generator does not imply ASP of the induced set-valued system (Zhao, 2023). For IFS, every point can be chain recurrent while ASP fails (Nia, 2015). In noncompact spaces, even the tight relationship between ASP and AASP breaks down: there are examples with ASP but not AASP, and with AASP but not ASP (Kulczycki et al., 2013).

A further limitation concerns equicontinuity. Since ASP implies $\{x_n\}_{n=0}^\infty$5-shadowing for every $\{x_n\}_{n=0}^\infty$6, and the density-shadowing results of the 2014 paper exclude $\{x_n\}_{n=0}^\infty$7- and $\{x_n\}_{n=0}^\infty$8-shadowing for nontrivial equicontinuous surjective systems, ASP is excluded from that class as well (Wu et al., 2014). Accordingly, the mature theory presents ASP as a distinctly non-equicontinuous phenomenon: weaker than pointwise shadowing and specification in some directions, but still incompatible with rigid distal behavior in many of its natural settings.