Shadowing Modulo an Ideal

• Shadowing modulo an ideal is an ideal-parameterized refinement of the pseudo-orbit tracing property, where a nonempty set from the ideal ensures accurate pseudo-orbits are traced by exact orbits.
• It extends classical shadowing by incorporating I-expansivity and I-topological stability, using the ideal to restrict the admissible subsets of the time semigroup.
• The framework utilizes unique trace detection, compactness-based lemmas, and continuity arguments to link perturbed actions to exact orbit traces.

Searching arXiv for papers on shadowing modulo an ideal and closely related shadowing variants. arxiv_search(query="shadowing modulo ideal dynamical systems", max_results=10, sort_by="relevance") arxiv_search(query="shadowing modulo an ideal", max_results=10) Shadowing modulo an ideal is an ideal-parameterized refinement of the pseudo orbit tracing property for transformation semigroups. In the formulation developed for uniform transformation semigroups, one fixes an ideal $\mathcal I$ on the phase semigroup $T$ and asks whether there exists a nonempty set $A\in\mathcal I$ such that every sufficiently accurate pseudo-orbit with respect to $A$ is traced by an exact orbit. The notion is introduced together with expansivity modulo an ideal and topological stability modulo an ideal for compact Hausdorff transformation semigroups, and classical shadowing appears as the special case of $\mathcal P_{fin}(T)$-shadowing (Shirazi et al., 24 Aug 2025).

1. Formal setting and ambient structures

The basic framework is a uniform transformation semigroup $(T,(X,\mathcal K),\rho)$. Here $T$ is a discrete topological semigroup with identity element $e$, $X$ is a topological space, and $\rho:T\times X\to X$ is a continuous action satisfying

$T$0

A compatible uniform structure $T$1 on $T$2 is fixed, so $T$3 is a uniform space. In the compact Hausdorff case, $T$4 has a unique compatible uniform structure, and the notation is often abbreviated to $T$5 with $T$6 denoting the action (Shirazi et al., 24 Aug 2025).

An ideal on a set $T$7 is a collection $T$8 such that $T$9, $A\in\mathcal I$0, and $A\in\mathcal I$1. In the dynamical setting, $A\in\mathcal I$2 is an ideal on $A\in\mathcal I$3. Typical examples are $A\in\mathcal I$4, $A\in\mathcal I$5, and $A\in\mathcal I$6. The paper does not impose translation invariance or other extra algebraic conditions on $A\in\mathcal I$7 (Shirazi et al., 24 Aug 2025).

This formalism differs from the single-map setting of classical shadowing theory, where one studies a continuous self-map $A\in\mathcal I$8 on a compact metric space. The semigroup formulation allows the time set to be an arbitrary discrete semigroup, while the ideal $A\in\mathcal I$9 specifies which subsets of $A$0 are admissible in the modified shadowing and expansivity conditions.

2. Pseudo-orbits, traces, and the definition of $A$1-shadowing

The key intermediate notion is shadowing with respect to a set $A$2. For a nonempty subset $A$3, a sequence $A$4 in $A$5, an entourage $A$6, and a point $A$7, one says that $A$8 is a $A$9-pseudo orbit with respect to $\mathcal P_{fin}(T)$0 if

$\mathcal P_{fin}(T)$1

One says that $\mathcal P_{fin}(T)$2 is a $\mathcal P_{fin}(T)$3-trace of $\mathcal P_{fin}(T)$4 if

$\mathcal P_{fin}(T)$5

A uniform transformation semigroup has shadowing property with respect to $\mathcal P_{fin}(T)$6 if for each $\mathcal P_{fin}(T)$7 there exists $\mathcal P_{fin}(T)$8 such that every $\mathcal P_{fin}(T)$9-pseudo orbit with respect to $(T,(X,\mathcal K),\rho)$0 has an $(T,(X,\mathcal K),\rho)$1-trace. The ideal-modified notion is then defined by existence of such a set inside the ideal: $(T,(X,\mathcal K),\rho)$2 has shadowing property modulo ideal $(T,(X,\mathcal K),\rho)$3, or $(T,(X,\mathcal K),\rho)$4-shadowing property, if there exists nonempty $(T,(X,\mathcal K),\rho)$5 such that the system has shadowing property with respect to $(T,(X,\mathcal K),\rho)$6. The paper further says that $(T,(X,\mathcal K),\rho)$7 has shadowing property or pseudo orbit tracing property (POTP) if it has $(T,(X,\mathcal K),\rho)$8-shadowing property (Shirazi et al., 24 Aug 2025).

Several structural facts clarify how the set $(T,(X,\mathcal K),\rho)$9 functions. For compact $T$0, adding the identity $T$1 to $T$2 does not change the property. More generally, if $T$3 lies in the subsemigroup generated by $T$4, then shadowing with respect to $T$5 is equivalent to shadowing with respect to $T$6; in particular, if $T$7 is generated by one element $T$8 and $T$9 is finite, then shadowing with respect to $e$0 is equivalent to shadowing with respect to $e$1 (Shirazi et al., 24 Aug 2025). This recovers the familiar time-one viewpoint for $e$2- or $e$3-actions.

The associated corollary states that the following are equivalent: first, there exists $e$4 such that $e$5 has shadowing property with respect to $e$6; second, for each $e$7 there exists $e$8 such that $e$9 and $X$0 has shadowing property with respect to $X$1. This shows that $X$2-shadowing is stable under enlarging the controlling set inside the ideal (Shirazi et al., 24 Aug 2025).

Conceptually, the ideal $X$3 encodes a notion of “smallness” or “negligibility” for subsets of the time semigroup. The definition then says that one works with pseudo-orbit constraints only along some $X$4, rather than uniformly across all of $X$5.

3. Relation to classical shadowing and limit shadowing

In the classical setting of a continuous self-map $X$6 on a compact metric space $X$7, a sequence $X$8 is a $X$9-pseudo orbit if

$\rho:T\times X\to X$0

and $\rho:T\times X\to X$1 has the shadowing property if for any $\rho:T\times X\to X$2 there is $\rho:T\times X\to X$3 such that every $\rho:T\times X\to X$4-pseudo orbit is $\rho:T\times X\to X$5-shadowed by some point of $\rho:T\times X\to X$6. A limit pseudo orbit satisfies

$\rho:T\times X\to X$7

and $\rho:T\times X\to X$8 has the limit shadowing property if every limit pseudo orbit has a point $\rho:T\times X\to X$9 such that

$T$00

(Kawaguchi, 2017).

These are not the same as shadowing modulo an ideal. Limit shadowing weakens the tracing requirement in an asymptotic sense, whereas $T$01-shadowing changes which semigroup elements are used to define pseudo-orbits. Nonetheless, the comparison is useful because both theories refine classical shadowing by relaxing uniform time-by-time control.

A central theorem in the single-map literature states that if a continuous map has the limit shadowing property, then

$T$02

and the restriction $T$03 satisfies the shadowing property (Kawaguchi, 2017). For equicontinuous maps, the limit shadowing property, the shadowing property, and the condition $T$04 are equivalent (Kawaguchi, 2017). These results supply structural background for ideal-modified shadowing: they show that weaker tracing notions often become stronger after restriction to an invariant core or after imposing additional regularity such as equicontinuity.

The semigroup-based notion of $T$05-shadowing should therefore be read as a different direction of generalization. It is not asymptotic in the sense of limit shadowing; rather, it is indexed by an ideal on the time semigroup and is built to interact with ideal-modified expansivity and ideal-modified topological stability.

4. Expansivity modulo an ideal and topological stability modulo an ideal

The accompanying expansivity notion is defined as follows. A uniform transformation semigroup is expansive modulo ideal $T$06, or $T$07-expansive, if there exists $T$08 such that for all distinct $T$09 and $T$10, there exists $T$11 with $T$12. Equivalently, if $T$13 is an $T$14-expansive index, then

$T$15

Moreover, if $T$16, then $T$17-expansive implies $T$18-expansive, so $T$19-expansive implies classical expansive (Shirazi et al., 24 Aug 2025).

Topological stability modulo an ideal is defined on the action space. For $T$20,

$T$21

For $T$22 and $T$23,

$T$24

The ideal-dependent uniform structure on $T$25 is

$T$26

An action $T$27 is topological stable modulo ideal $T$28 if for each $T$29 there exists an open neighbourhood $T$30 of $T$31 in $T$32 such that for each $T$33 there exists a homomorphism

$T$34

with $T$35. Classical topological stability is recovered as $T$36-topological stability (Shirazi et al., 24 Aug 2025).

The main stability theorem states that every $T$37-expansive, compact Hausdorff transformation (semi)group with $T$38-shadowing property is $T$39-topological stable (Shirazi et al., 24 Aug 2025). This is the ideal-modified analogue of the classical implication “shadowing + expansivity $T$40 topological stability.”

The topology determined by $T$41 interpolates between standard regimes. If $T$42, then $T$43 is the product, or pointwise convergence, uniformity on $T$44. If $T$45, then $T$46 is the uniform convergence topology on $T$47. If $T$48, the topology is trivial (Shirazi et al., 24 Aug 2025).

5. Proof architecture and core lemmas

The proof of the ideal-modified stability theorem follows the same broad pattern as classical Walters-type arguments, but every step is reformulated in ideal-sensitive language.

The first ingredient is uniqueness of traces under $T$49-expansivity. If $T$50 is $T$51-expansive with index $T$52, and $T$53 satisfies $T$54, then any sequence $T$55 has at most one $T$56-trace. Indeed, if $T$57 and $T$58 are both $T$59-traces, then $T$60 for all $T$61, contradicting $T$62-expansivity unless $T$63 (Shirazi et al., 24 Aug 2025).

The second ingredient combines expansivity and shadowing with respect to a fixed set $T$64. If the system is $T$65-expansive with index $T$66 and has shadowing with respect to $T$67, then there exist $T$68 with $T$69 such that every $T$70-pseudo orbit with respect to $T$71 has a unique $T$72-trace (Shirazi et al., 24 Aug 2025).

A third ingredient is a compactness-based finite detection lemma. If $T$73 is compact and the system is $T$74-expansive with closed index $T$75, then for every open entourage $T$76 and every $T$77, there exists a nonempty finite subset $T$78 such that

$T$79

where $T$80. Thus closeness along finitely many times outside an ideal-small exceptional set forces closeness at time $T$81 (Shirazi et al., 24 Aug 2025).

The bridge from nearby actions to pseudo-orbits is provided by the statement that if $T$82, then for every $T$83 the sequence $T$84 is a $T$85-pseudo orbit with respect to $T$86 in $T$87. This converts perturbations of the action into pseudo-orbits for the original system (Shirazi et al., 24 Aug 2025).

The proof of topological stability then proceeds by choosing $T$88 witnessing $T$89-shadowing, selecting an entourage that guarantees unique traces, and taking a $T$90-small neighbourhood $T$91 of the original action. For each perturbed action $T$92 and each $T$93, the orbit $T$94 has a unique trace $T$95 under the original action, and this defines a map $T$96. One then shows that $T$97 satisfies

$T$98

so $T$99 is a homomorphism from the perturbed action to the original one, and that $A\in\mathcal I$00 is uniformly close to $A\in\mathcal I$01 and continuous (Shirazi et al., 24 Aug 2025). The conclusion is precisely $A\in\mathcal I$02-topological stability.

6. Examples, counterexamples, and terminological scope

A principal counterexample shows that $A\in\mathcal I$03-shadowing can hold even when classical shadowing fails. Let

$A\in\mathcal I$04

with the metric induced from $A\in\mathcal I$05, and define $A\in\mathcal I$06 by

$A\in\mathcal I$07

With $A\in\mathcal I$08, the transformation group $A\in\mathcal I$09 does not have shadowing property (POTP) but does have $A\in\mathcal I$10-shadowing property (Shirazi et al., 24 Aug 2025). This example establishes that ideal-modified shadowing is genuinely weaker than classical POTP for suitable ideals.

A second counterexample distinguishes classical expansivity from $A\in\mathcal I$11-expansivity. Let $A\in\mathcal I$12 with the Euclidean metric, define $A\in\mathcal I$13 for $A\in\mathcal I$14, let

$A\in\mathcal I$15

and consider the induced transformation semigroup. Then $A\in\mathcal I$16 is expansive classically but not $A\in\mathcal I$17-expansive (Shirazi et al., 24 Aug 2025). Thus the ideal-modified expansivity condition is not a mere reformulation of standard expansivity.

For discrete $A\in\mathcal I$18 with $A\in\mathcal I$19, $A\in\mathcal I$20-topological stability is characterized by isolation in the action space: an action $A\in\mathcal I$21 is $A\in\mathcal I$22-topological stable if and only if $A\in\mathcal I$23 is an isolated point of $A\in\mathcal I$24 with respect to the topology induced by $A\in\mathcal I$25 (Shirazi et al., 24 Aug 2025). This identifies an especially rigid regime in which stability becomes a local topological property of the action space itself.

The phrase “shadowing modulo an ideal” also appears in a different, non-dynamical sense in arithmetic distance geometry. In the setting of finite point sets $A\in\mathcal I$26, one paper studies the situation where the squared distance set $A\in\mathcal I$27 lies in $A\in\mathcal I$28 and occupies only finitely many nonzero residue classes modulo a prime ideal $A\in\mathcal I$29; the paper describes this as a modular “shadow” of the full distance structure and proves the bound

$A\in\mathcal I$30

under the stated congruence hypothesis (Nozaki, 2022). This usage concerns a sparse residue-class image of geometric distances rather than pseudo-orbit tracing. A plausible implication is that the expression “shadowing modulo an ideal” is terminologically ambiguous across subfields: in dynamical systems it refers to ideal-constrained tracing properties of orbits, whereas in the arithmetic-geometric context it refers to reduction of distance data modulo a prime ideal.