Almost Average Shadowing Property

• The almost average shadowing property (ALASP) is defined so that every pseudo-orbit with a small limsup Cesàro error is approximated by a true orbit, refining conventional shadowing schemes.
• ALASP bridges strict pseudo-orbit tracing and average-based methods, providing a weaker admissibility condition that implies the standard average shadowing property in surjective systems.
• Under compactness and stability conditions, ALASP yields strong dynamical consequences such as chain transitivity, mixing, and a unique chain component, highlighting its role in the global connectivity of dynamics.

Searching arXiv for the cited papers and closely related work on almost average shadowing. Almost Average Shadowing Property, usually abbreviated ALASP, is a shadowing notion for a continuous self-map $f:X\to X$ on a metric space $(X,d)$. In the formulation introduced by Mukta Garg and Ruchi Das, ALASP requires that every sequence whose one-step errors are small in the limsup Cesàro sense be traced, again in the limsup Cesàro sense, by a genuine orbit: for every $\varepsilon>0$ there exists $\delta>0$ such that any sequence $\{x_i\}_{i\ge 0}$ with

$$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$

is traced by some $x\in X$ satisfying

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

This places ALASP between strict pseudo-orbit tracing and weaker density- or average-based shadowing schemes, and also explains why the notion is closely entangled with average chain transitivity, average chain mixing, and asymptotic average shadowing (Garg et al., 2016).

1. Formal definition and immediate comparison with average shadowing

The ambient setting is a dynamical system $(X,f)$, where $X$ is a metric space with metric $(X,d)$0 and $(X,d)$1 is continuous. Garg and Das define an almost $(X,d)$2-average-pseudo-orbit to be a sequence $(X,d)$3 such that

$(X,d)$4

Such a sequence is $(X,d)$5-shadowed in average by a point $(X,d)$6 when

$(X,d)$7

The map $(X,d)$8 has the Almost Average Shadowing Property if for every $(X,d)$9 there exists $\varepsilon>0$0 such that every almost $\varepsilon>0$1-average-pseudo-orbit is $\varepsilon>0$2-shadowed in average by some point of $\varepsilon>0$3 (Garg et al., 2016).

In the same paper, the standard average-shadowing property (ASP) is stated using a stronger pseudo-orbit hypothesis. A sequence $\varepsilon>0$4 is a $\varepsilon>0$5-average-pseudo-orbit if there exists $\varepsilon>0$6 such that for all $\varepsilon>0$7 and all $\varepsilon>0$8,

$\varepsilon>0$9

Thus ALASP weakens the admissibility condition from uniform control over all sufficiently long sliding windows to a single limsup condition over initial segments. Because every $\delta>0$0-average-pseudo-orbit is, in particular, an almost $\delta>0$1-average-pseudo-orbit, ALASP implies ASP (Garg et al., 2016).

This distinction is central. ALASP is not defined by uniform coordinatewise control, nor by a density-one tracing requirement, but by Cesàro control of local defect and Cesàro control of tracing error. The adjective “almost” modifies the pseudo-orbit condition rather than the tracing clause in the original Garg–Das formulation (Garg et al., 2016).

2. Relation to neighboring notions and terminological variation

The literature surrounding average shadowing is terminologically non-uniform. Several nearby notions coexist, and different papers use “almost” in different ways. The following table summarizes the principal average-type notions that appear in the cited works.

Notion	Pseudo-orbit condition	Tracing condition
ALASP	$\delta>0$2	$\delta>0$3
ASP	sliding-window average error $\delta>0$4	$\delta>0$5
AASP	$\delta>0$6	$\delta>0$7
AAASP	weak asymptotic average pseudo-orbit with limsup bound $\delta>0$8	average tracing with limsup bound $\delta>0$9

In Kulczycki–Kwietniak–Oprocha, the exact term “almost average shadowing property” is not introduced. That paper instead develops the relations among almost specification, asymptotic average shadowing property (AAvSh or AASP), and average shadowing property (AvSh), proving on compact surjective systems that

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

It also records that under the classical shadowing property these average-type properties become equivalent to strong transitivity and specification conditions (Kulczycki et al., 2013).

A similar terminological caution appears in Kwietniak–Łącka–Oprocha. There, the operative notion is again AASP, not ALASP, and the main measure-theoretic consequence is that AASP implies that every invariant measure has a generic point. That paper also proves that weak specification implies AASP and uses the Besicovitch pseudometric $\{x_i\}_{i\ge 0}$1 as the main technical device (Kwietniak et al., 2016).

Later work makes the ambiguity explicit. In the gluing-based framework of “Average shadowing revisited,” the label ALASP is not used as a formal term, but the paper identifies what it calls “almost average shadowing” with the strong-average property $\{x_i\}_{i\ge 0}$2, whereas ASP is identified with $\{x_i\}_{i\ge 0}$3, where $\{x_i\}_{i\ge 0}$4 denotes strong average pseudo-orbits and $\{x_i\}_{i\ge 0}$5 weak average pseudo-orbits. In that notation, ASP implies ALASP because $\{x_i\}_{i\ge 0}$6 (Blank, 2022). This suggests that the phrase “almost average shadowing property” has acquired more than one technical meaning across adjacent strands of the literature.

3. Average chains and global dynamical consequences

Garg and Das introduced ALASP together with a family of average chain notions. For $\{x_i\}_{i\ge 0}$7 and $\{x_i\}_{i\ge 0}$8, a $\{x_i\}_{i\ge 0}$9-average-chain from $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$0 to $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$1 of length $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$2 is a finite sequence

$$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$3

for which there exists $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$4, $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$5, such that for all $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$6,

$$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$7

The map $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$8 is average chain transitive if for any $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f(x_i),x_{i+1}\big)<\delta$$9 and any $x\in X$0 there exists a $x\in X$1-average-chain from $x\in X$2 to $x\in X$3. It is average chain mixing if for any $x\in X$4 and any $x\in X$5 there exists $x\in X$6 such that for every $x\in X$7 there is a $x\in X$8-average-chain from $x\in X$9 to $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$0 of length $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$1 (Garg et al., 2016).

The elementary implications recorded in that paper are

$$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$2

and

$$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$3

These implications place average-chain notions below their classical chain analogues (Garg et al., 2016).

ALASP has substantially stronger consequences than average chain transitivity alone. If $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$4 is compact and $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$5 is surjective, then ALASP implies that $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$6 is chain transitive; in particular, the chain recurrent set satisfies $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$7. Under the same compact surjective hypotheses, ALASP also implies chain mixing. Since ALASP is preserved by iterates, the proof of chain mixing passes through total chain transitivity and the known equivalence between total chain transitivity and chain mixing on compact metric spaces (Garg et al., 2016).

Further, on a compact dynamical system, ALASP forces the chain recurrent set to consist of a single chain component. This excludes decomposition of $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$8 into distinct chain components and shows that ALASP imposes a strong coarse connectivity on the dynamics (Garg et al., 2016).

A separate recurrence consequence appears when minimal points are dense. If $$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d\big(f^i(x),x_i\big)<\varepsilon.$$9 is compact, $(X,f)$0 has ALASP, and the minimal points of $(X,f)$1 are dense in $(X,f)$2, then $(X,f)$3 is totally strongly ergodic (Garg et al., 2016).

4. Stability properties and permanence under constructions

One of the notable features of ALASP in the Garg–Das framework is its permanence under several standard operations. If $(X,f)$4 has ALASP, then every iterate $(X,f)$5, $(X,f)$6, also has ALASP. The proof is based on converting an almost $(X,f)$7-average-pseudo-orbit for $(X,f)$8 into an almost $(X,f)$9-average-pseudo-orbit for $X$0 by inserting the intermediate points $X$1 between successive $X$2 (Garg et al., 2016).

ALASP is also stable under bounded products. If $X$3 and $X$4 are bounded dynamical systems and both $X$5 and $X$6 have ALASP, then $X$7 has ALASP on $X$8 with the product metric

$X$9

The proof uses explicit upper-density estimates for the set of times at which one of the coordinatewise tracing errors exceeds a prescribed threshold (Garg et al., 2016).

The surrounding average-chain theory provides additional permanence statements. If $(X,d)$00 is average chain transitive for some $(X,d)$01, then $(X,d)$02 is average chain transitive. If $(X,d)$03 is Lipschitz and average chain mixing, then $(X,d)$04 is totally average chain transitive. Moreover, average chain mixing implies that $(X,d)$05 is average chain transitive, and the same conclusion holds when $(X,d)$06 is totally average chain transitive (Garg et al., 2016). These results are not themselves ALASP theorems, but they clarify the average-chain background in which ALASP was introduced.

5. Examples, counterexamples, and separations

The basic positive example is the class of constant maps. Garg and Das note that constant maps have ALASP: every pseudo-orbit, whether average or almost average, is followed by the constant orbit in the required averaged sense (Garg et al., 2016).

The simplest negative example is the two-point discrete system $(X,d)$07 with the permutation $(X,d)$08, $(X,d)$09. For every $(X,d)$10, Garg and Das construct an almost $(X,d)$11-average-pseudo-orbit that cannot be $(X,d)$12-shadowed in average for $(X,d)$13. Hence this two-cycle does not have ALASP (Garg et al., 2016).

ALASP is strictly stronger than ASP. Garg and Das record that the space $(X,d)$14 and map $(X,d)$15 from the noncompact examples of Kulczycki–Kwietniak–Oprocha have ASP but do not have ALASP. The same source also shows that compactness is essential for several implications among average-shadowing notions [(Garg et al., 2016); (Kulczycki et al., 2013)].

Classical shadowing does not imply ALASP. On the Cantor set $(X,d)$16 with its usual Cantor metric, the identity map has the shadowing property, but it does not have ALASP. In the argument reported by Garg and Das, every point is minimal and minimal points are dense, so Theorem 4.8 excludes ALASP (Garg et al., 2016).

Average chain mixing is also strictly weaker than ALASP. The same paper gives examples of systems that are average chain mixing but not chain transitive: a constant map on a metric space with more than one element, the identity map on a union of two disjoint circles, and the map $(X,d)$17 on the finite discrete space $(X,d)$18. The authors explicitly note that the two-point 2-cycle shows that average chain mixing need not imply ALASP (Garg et al., 2016).

6. Extensions, measure-theoretic links, and later equivalence results

Although ALASP itself was introduced in (Garg et al., 2016), most later developments in the arXiv literature focus on adjacent notions such as ASP, AASP, mean ergodic shadowing, and density-based shadowing. In “On various definitions of shadowing with average error in tracing,” Kulczycki, Oprocha, and collaborators proved that almost specification implies AASP and that AASP implies ASP, without assuming surjectivity. They also established equivalent formulations of ASP in terms of $(X,d)$19-shadowing for every $(X,d)$20, weak asymptotic average shadowing, and shadowing of $(X,d)$21-asymptotic-average pseudo-orbits (Wu et al., 2014).

The measure-theoretic significance of asymptotic average shadowing was sharpened in “Generic points for dynamical systems with average shadowing.” For compact dynamical systems, every invariant measure can be realized by an asymptotic average pseudo-orbit, and any point asymptotically tracing that pseudo-orbit in average is generic for the measure. Consequently, AASP implies that every invariant measure has a generic point. The proof uses continuity of the distribution-measures map with respect to the Besicovitch pseudometric $(X,d)$22, and it also shows that the set of generic points of ergodic measures is $(X,d)$23-closed (Kwietniak et al., 2016).

Gluing-based frameworks broaden the scope of average shadowing beyond continuous invertible maps. In the work of Medvedev and coauthors, the key structural assumption is a gluing property with summable rate function. Under strong gluing with $(X,d)$24, the system has average shadowing for weak-average pseudo-orbits, and hence also the stronger “almost average” behavior in the sense $(X,d)$25. These papers treat discontinuous and non-invertible systems and emphasize perturbations that are small only on average rather than uniformly (Blank, 2022, Blank, 2022).

A further development concerns almost asymptotic average shadowing rather than ALASP in the Garg–Das sense. Das and Bag proved for finitely generated free semigroup actions that the average shadowing property, weak asymptotic average shadowing property, mean ergodic shadowing property, almost asymptotic average shadowing property, asymptotic average shadowing property, and $(X,d)$26-shadowing property for every $(X,d)$27 are equivalent. In the autonomous case, the same equivalence holds for continuous maps on compact metric spaces, giving an affirmative answer to the question of whether ASP implies AASP (Das et al., 10 Apr 2025).

Taken together, these results show that ALASP belongs to a broader hierarchy of Cesàro-type tracing properties. In the original definition, it strengthens ASP by weakening only the pseudo-orbit admissibility condition; in later usage, the phrase may instead denote strong-average shadowing in a gluing framework. This suggests that precise identification of the pseudo-orbit class and tracing criterion is indispensable whenever “almost average shadowing” is invoked.