Asymptotic Average Shadowing Property

• AASP is a dynamical property where pseudo-orbits are traced on average, requiring the mean deviation between the orbit and pseudo-orbit to vanish asymptotically.
• It employs tools like the Besicovitch pseudometric to compare orbits, establishing links with shadowing, specification, and mixing properties in various systems.
• The property extends to flows, nonautonomous systems, and semigroup actions, influencing attractor structures and the statistical regularity of invariant measures.

The asymptotic average shadowing property (AASP) is a shadowing property for dynamical systems in which the local errors of a pseudo-orbit are required to vanish only in average, and the tracing orbit is required to match that pseudo-orbit only in average as well. For a continuous map $f:X\to X$ on a compact metric space $(X,d)$, a sequence $\{x_i\}_{i=0}^\infty$ is an asymptotic average pseudo-orbit when

$$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$

and $f$ has AASP when every such sequence is asymptotically shadowed in average by some $z\in X$, meaning

$$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$

The notion is attributed to Gu, and one treatment explicitly attributes it to Gu and Xia; it occupies a central place in the modern hierarchy of shadowing and specification-like properties for maps, homeomorphisms, flows, iterated function systems, and related nonautonomous or semigroup actions (Wu et al., 2014, Carvalho et al., 2014).

1. Formal definitions and basic formulations

In the standard discrete-time setting, AASP is an asymptotic version of average shadowing. Ordinary shadowing asks for uniform $\varepsilon$-tracing of every sufficiently accurate pseudo-orbit, while AASP asks only for vanishing average tracing error along pseudo-orbits whose own one-step errors have vanishing average. This distinction is explicit in the definitions used in the shadowing literature on compact metric systems (Kwietniak et al., 2016, Wu et al., 2014).

A useful reformulation uses the Besicovitch pseudometric. For $(X,T)$, one writes

$$\rho_B(x,z)=\limsup_{n\to\infty}\frac1n\sum_{i=0}^{n-1}\rho(x_i,z_i),$$

and, for a point $(X,d)$0, denotes its orbit sequence by $(X,d)$1. In this language, $(X,d)$2 has AASP if for every asymptotic average pseudo-orbit $(X,d)$3, there exists $(X,d)$4 such that $(X,d)$5. The same framework supports equivalent density formulations and a comparison with the density-type pseudometric $(X,d)$6, which is uniformly equivalent to $(X,d)$7 (Can et al., 2024).

For flows, the definition is modified by reparametrization. In the geometric Lorenz and conservative-flow settings, an asymptotic average pseudo-orbit is a bi-sequence $(X,d)$8 with $(X,d)$9 and

$\{x_i\}_{i=0}^\infty$0

or a one-sided analogue, while asymptotic average shadowing is expressed through averaged integral discrepancies along orbit pieces and an orientation-preserving time change $\{x_i\}_{i=0}^\infty$1, or through $\{x_i\}_{i=0}^\infty$2 in the bounded-distortion formulation (Arbieto et al., 2013, Bessa et al., 2013).

The same pattern extends beyond autonomous maps. For nonautonomous systems $\{x_i\}_{i=0}^\infty$3, one replaces $\{x_i\}_{i=0}^\infty$4 by the cocycle iterates $\{x_i\}_{i=0}^\infty$5; for parameterized IFS one replaces $\{x_i\}_{i=0}^\infty$6 by symbol-dependent compositions; and for finitely generated free semigroup actions one works along a fixed generator sequence $\{x_i\}_{i=0}^\infty$7 and the corresponding compositions $\{x_i\}_{i=0}^\infty$8 (An, 2024, Nia, 2015, Das et al., 10 Apr 2025).

2. Position in the hierarchy of shadowing and specification properties

AASP sits inside a rich implication structure. One systematic treatment proved

$\{x_i\}_{i=0}^\infty$9

and also showed that AASP implies the density-based shadowing properties usually denoted $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$0-shadowing and $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$1-shadowing. In that work, almost specification implies AASP without any surjectivity assumption (Wu et al., 2014).

Another line of results places AASP below stronger pseudo-orbit tracing notions. For homeomorphisms of compact metric spaces, two-sided limit shadowing implies shadowing and transitivity; combining this with the Kulczycki–Kwietniak–Oprocha equivalence theorem yields topological mixing, specification, average shadowing, and asymptotic average shadowing. Thus two-sided limit shadowing implies AASP, whereas the weaker “two-sided limit shadowing with a gap” implies only shadowing and transitivity, not AASP in the stated results (Carvalho et al., 2014).

Weak specification also implies AASP. In the Besicovitch framework, this implication is proved for surjective systems, and it is accompanied by the statement that AASP is strictly weaker than weak specification and also weaker than almost specification in general. Later work sharpened the picture further by proving that vague specification is equivalent to AASP, and that weak specification implies vague specification while the converse fails (Kwietniak et al., 2016, Can et al., 2024).

The relation between average shadowing and AASP has evolved substantially. Earlier compact-system work established AASP $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$2 average shadowing and posed the converse as an open problem, while later results proved, for finitely generated free semigroup actions, that average shadowing, weak asymptotic average shadowing, mean ergodic shadowing, almost asymptotic average shadowing, AASP, and $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$3-shadowing for every $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$4 are equivalent. The autonomous single-map case appears as a special case of that theorem, giving an affirmative answer to the question posed in 2014 (Kulczycki et al., 2013, Das et al., 10 Apr 2025). In a different direction, one 2024 result shows that for surjective systems AASP is equivalent to average shadowing provided the phase space is complete with respect to the dynamical Besicovitch pseudometric (Can et al., 2024).

3. Dynamical consequences

AASP has strong consequences for the statistical structure of a system. A central theorem in the Besicovitch approach states that for every invariant measure $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$5 one can build an asymptotic average pseudo-orbit such that any point asymptotically tracing it in average is generic for $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$6. Consequently, every invariant measure has a generic point in any system with AASP. The same work proves that the set of generic points of ergodic measures is closed with respect to the Besicovitch pseudometric, and that every distribution measure of an asymptotic average pseudo-orbit is invariant (Kwietniak et al., 2016).

In the presence of shadowing, AASP is tied to strong recurrence and mixing. For homeomorphisms with shadowing, total transitivity, topological mixing, average shadowing, AASP, and specification are equivalent in the Kulczycki–Kwietniak–Oprocha theorem used repeatedly in later papers. One consequence is that, in settings where shadowing is already available, AASP is often indistinguishable from strong orbit-gluing properties and mixing behavior (Carvalho et al., 2014).

For flows on isolated invariant sets, AASP also constrains attractor structure. A direct result for $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$7-vector fields states that if an isolated set $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$8 has the asymptotic average shadowing property, then $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f(x_i),x_{i+1}\bigr)=0,$$9 has no proper attractor. The proof constructs a bi-infinite asymptotic average pseudo-orbit that follows one orbit on negative indices and another orbit outside the basin of a putative attractor on positive indices, contradicting asymptotic average tracing (Chu et al., 2016).

AASP is also inherited by factors. This follows from the equivalence between AASP and vague specification together with the fact that vague specification is inherited by factors. The factor-inheritance statement answers one of the questions raised in the earlier literature on AASP (Can et al., 2024).

4. Principal classes of systems with AASP

Several natural dynamical classes are known to carry AASP. For homeomorphisms, two-sided limit shadowing is among the strongest known tracing properties and implies AASP. In expansive settings, this yields especially sharp characterizations: for expansive homeomorphisms, transitivity plus shadowing is equivalent to two-sided limit shadowing with a gap, and mixing plus shadowing is equivalent to two-sided limit shadowing. For shift spaces, a transitive shift of finite type is characterized by two-sided limit shadowing with a gap, and a topologically mixing shift of finite type is characterized by two-sided limit shadowing itself (Carvalho et al., 2014).

AASP is not confined to expansive systems. The full shift $f$0 has two-sided limit shadowing for any compact metric space $f$1, so when $f$2 one obtains a non-expansive homeomorphism with shadowing, average shadowing, AASP, and specification. The same paper also constructs products of systems with two-sided limit shadowing and shows that the product retains that property (Carvalho et al., 2014).

Specification-like properties supply another large source of examples. Weak specification implies AASP, and almost specification implies AASP. A later equivalence theorem with vague specification then produces examples that are much weaker than classical specification behavior: certain proximal and minimal shift spaces possess vague specification, hence AASP. This resolves earlier questions asking whether nontrivial minimal systems with AASP exist (Kwietniak et al., 2016, Wu et al., 2014, Can et al., 2024).

Genericity results show that AASP is abundant in conservative topological dynamics. For compact connected manifolds, special shadowing is generic in $f$3, and because generic conservative homeomorphisms are topologically mixing while mixing plus shadowing implies average shadowing and asymptotic average shadowing, a generic element of $f$4 satisfies specification, average shadowing, and AASP (Guihéneuf et al., 2016).

5. Extensions beyond a single autonomous map

For parameterized iterated function systems $f$5, AASP is defined by requiring that there exist symbol sequences $f$6 such that

$f$7

and that some symbol-driven orbit asymptotically shadows the pseudo-orbit in average. Every uniformly contracting IFS has AASP, AASP is invariant under topological conjugacy, $f$8 has AASP if and only if $f$9 has AASP, and the product $z\in X$0 has AASP if and only if both factors do. If each branch is surjective and the phase space is compact, AASP implies chain transitivity. The paper also gives an explicit IFS with AASP but without the ordinary shadowing property (Nia, 2015).

For nonautonomous systems $z\in X$1, AASP is defined in the same average-asymptotic manner using the cocycle iterates $z\in X$2. A transfer theorem states that if a closed invariant subset $z\in X$3 has AASP under the restricted system and every orbit spends a proportion $z\in X$4 of its first $z\in X$5 iterates $z\in X$6-close to $z\in X$7, uniformly in the initial point, then the whole system has AASP. The proof uses density-zero exceptional sets and an equivalence between vanishing average and pointwise convergence off a set of density zero (An, 2024).

For finitely generated free semigroup actions, the asymptotic average shadowing property is defined relative to a generator sequence $z\in X$8. In this setting, AASP is equivalent to the average shadowing property, the weak asymptotic average shadowing property, the mean ergodic shadowing property, the almost asymptotic average shadowing property, and $z\in X$9-shadowing for every $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$0. This equivalence specializes to ordinary continuous maps on compact metric spaces (Das et al., 10 Apr 2025).

A nearby but distinct development appears in the semi-hyperbolic literature on “asymptotic bi-shadowing.” That theory uses an exponential rate condition,

$$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$1

rather than an average condition, and is therefore not a formulation of classical AASP (Cheng et al., 30 Mar 2026).

6. Obstructions, counterexamples, and structural limits

AASP has sharp limitations in smooth and singular settings. For geometric Lorenz flows, if the induced map on the foliation satisfies

$$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$2

then the flow has neither $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$3-average shadowing, nor $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$4-limit shadowing, nor $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$5-AASP for any $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$6. The obstruction is established by constructing a bi-infinite asymptotic average pseudo-orbit near the singularity that no reparametrized orbit can shadow in the required averaged sense (Arbieto et al., 2013).

Robust AASP can also force strong hyperbolic structure. For incompressible flows, $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$7-stable average shadowing or $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$8-stable asymptotic average shadowing implies a dominated splitting on the whole manifold; in the Hamiltonian setting, $$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1} d\bigl(f^i(z),x_i\bigr)=0.$$9-robust asymptotic average shadowing implies partial hyperbolicity on the regular energy surface. By contrast, robust limit shadowing implies Anosov behavior in the corresponding conservative classes (Bessa et al., 2013).

The relation between AASP and ordinary shadowing is nontrivial. The IFS literature provides an explicit example with AASP but without shadowing (Nia, 2015). Conversely, the older compact-map literature emphasized that AASP and average shadowing differ from ordinary shadowing because they require only average, not uniform, closeness (Kwietniak et al., 2016). For compact maps, the later 2025 equivalence theorem collapses several average-type notions together, but this does not identify AASP with classical shadowing (Das et al., 10 Apr 2025).

Compactness is essential in several classical implication theorems. Earlier work exhibited noncompact systems $\varepsilon$0 with average shadowing but not AASP and $\varepsilon$1 with AASP but not average shadowing, showing that the compact hypotheses in the compact-metric-space theory are structurally necessary. This gives a natural boundary for the strongest equivalence statements (Kulczycki et al., 2013).

A final common misconception concerns the relation with stronger specification properties. AASP is not equivalent to weak specification or almost specification in general. The equivalence with vague specification, together with examples of proximal and minimal shift spaces possessing vague specification, shows that AASP can hold in systems far weaker than classical specification-like models (Can et al., 2024).