Topological Stability Modulo an Ideal

• Topological Stability Modulo an Ideal is a framework that refines classical dynamics by measuring perturbations with respect to negligible subsets defined by an ideal.
• It establishes that systems with ideal-relative shadowing and expansivity admit close homomorphisms, extending classic stability results.
• The approach unifies stability concepts across dynamical systems, topology, and algebra by systematically ignoring small, defect-prone subsets.

Searching arXiv for papers on topological stability modulo an ideal and closely related “modulo an ideal” frameworks. Topological stability modulo an ideal is, in its most explicit current formulation, a dynamical property of a uniform transformation semigroup $(T,X,\mathfrak X)$ relative to an ideal $\mathcal I$ on the time semigroup $T$: perturbations of the action are measured in a uniform structure indexed by $\mathcal I$, and sufficiently small perturbations admit a homomorphism close to the identity (Shirazi et al., 24 Aug 2025). More broadly, the phrase belongs to a family of “modulo an ideal” constructions in which an ideal specifies negligible subsets—of time, of space, or of an algebraic filtration—and classical notions such as connectedness, chaos, and asymptotic regularity are weakened by ignoring defects on those small sets (Koushesh, 2014, Pourattar et al., 2018).

1. Ideal-relative smallness as the organizing principle

The common mechanism is the replacement of absolute statements by relative ones. In the dynamical setting, an ideal $\mathcal I$ on a set $M$ is a nonempty family of subsets of $M$ such that if $A,B\in\mathcal I$, then $A\cup B\in\mathcal I$, and if $A\in\mathcal I$ and $\mathcal I$0, then $\mathcal I$1. Thus $\mathcal I$2 encodes the subsets regarded as “small”; typical examples are $\mathcal I$3, $\mathcal I$4, and $\mathcal I$5 (Shirazi et al., 24 Aug 2025).

An analogous set-theoretic notion is used in topology. For a space $\mathcal I$6, an ideal $\mathcal I$7 is likewise downward closed and closed under finite unions, and it supports the definition of connectedness modulo $\mathcal I$8. Here the ideal does not live on a time semigroup but on the underlying space itself (Koushesh, 2014).

A different but related use of “ideal” occurs in commutative algebra. For a commutative augmented ring $\mathcal I$9, the augmentation ideal is

$T$0

and the filtration

$T$1

produces successive quotients $T$2. In that setting, “modulo an ideal” refers to behavior relative to powers of a ring ideal rather than to a family of negligible subsets (Chang, 2019).

A persistent source of ambiguity is therefore terminological rather than mathematical: the phrase “modulo an ideal” does not refer to a single universal construction. In one line of work, the ideal is a family of negligible subsets of a set or semigroup; in another, it is an ordinary algebraic ideal that induces an adic topology and a graded filtration. The literature considered here uses both meanings, and the exact interpretation depends on context.

2. Uniform transformation semigroups and the formal definition of topological stability modulo an ideal

In the dynamical theory, a transformation semigroup consists of a discrete semigroup $T$3 with identity $T$4, a topological space $T$5, and a continuous action

$T$6

satisfying $T$7 and $T$8. When $T$9 carries a compatible uniform structure $\mathcal I$0, one writes $\mathcal I$1 (Shirazi et al., 24 Aug 2025).

The shadowing side of the theory is defined relative to a subset $\mathcal I$2. A sequence $\mathcal I$3 is a $\mathcal I$4-pseudo orbit with respect to $\mathcal I$5 if

$\mathcal I$6

and a point $\mathcal I$7 is a $\mathcal I$8-trace of $\mathcal I$9 if

$\mathcal I$0

The system has shadowing with respect to $\mathcal I$1 if every sufficiently fine pseudo orbit with respect to $\mathcal I$2 admits a trace. It has shadowing property modulo ideal $\mathcal I$3 if there exists a nonempty $\mathcal I$4 such that shadowing holds with respect to $\mathcal I$5 (Shirazi et al., 24 Aug 2025).

Expansivity is likewise ideal-relative. The system is expansive modulo $\mathcal I$6 if there exists $\mathcal I$7 such that for all distinct $\mathcal I$8 and all $\mathcal I$9, there exists

$M$0

Equivalently, for distinct $M$1,

$M$2

for an $M$3-expansive index $M$4 (Shirazi et al., 24 Aug 2025).

Topological stability modulo $M$5 is defined on the action space itself. For $M$6 and $M$7, one sets

$M$8

where $M$9 is the uniform-convergence entourage on $M$0. Then

$M$1

An action $M$2 is topologically stable modulo $M$3 if for each $M$4 there exists an open neighborhood $M$5 of $M$6 in $M$7 such that each $M$8 admits a homomorphism

$M$9

with $A,B\in\mathcal I$0 (Shirazi et al., 24 Aug 2025).

The central theorem states that every $A,B\in\mathcal I$1-expansive compact Hausdorff transformation (semi)group with $A,B\in\mathcal I$2-shadowing property is $A,B\in\mathcal I$3-topologically stable. This is the direct ideal-relative extension of the classical implication “shadowing $A,B\in\mathcal I$4 expansivity $A,B\in\mathcal I$5 topological stability” (Shirazi et al., 24 Aug 2025).

3. Relation to classical shadowing, expansivity, and perturbation theory

Classical notions are recovered by choosing the ideal of finite subsets. In particular, the usual shadowing property is exactly $A,B\in\mathcal I$6-shadowing, and classical topological stability is exactly $A,B\in\mathcal I$7-topological stability (Shirazi et al., 24 Aug 2025).

The formalism is not merely cosmetic. The paper gives a counterexample in which ideal-relative shadowing is strictly weaker than classical shadowing. Let

$A,B\in\mathcal I$8

let $A,B\in\mathcal I$9, and let $A\cup B\in\mathcal I$0, the ideal of subsets of odd integers. For the transformation group generated by the explicitly defined map $A\cup B\in\mathcal I$1, classical shadowing fails, but $A\cup B\in\mathcal I$2-shadowing holds because shadowing with respect to $A\cup B\in\mathcal I$3 can still be verified (Shirazi et al., 24 Aug 2025).

The same separation occurs for expansivity. On $A\cup B\in\mathcal I$4, with the semigroup $A\cup B\in\mathcal I$5 where $A\cup B\in\mathcal I$6, the action is classically expansive, but it is not $A\cup B\in\mathcal I$7-expansive for the ideal

$A\cup B\in\mathcal I$8

This shows that ideal-relative expansivity is not a monotone weakening of classical expansivity in any naive sense; it depends on where the separating times are allowed to lie (Shirazi et al., 24 Aug 2025).

Several structural facts clarify how the theory behaves. If $A\cup B\in\mathcal I$9 is compact and $A\in\mathcal I$0 belongs to the subsemigroup generated by a nonempty set $A\in\mathcal I$1, then shadowing with respect to $A\in\mathcal I$2 is equivalent to shadowing with respect to $A\in\mathcal I$3. In finitely generated compact systems this reduces shadowing with respect to a finite generating set to shadowing with respect to a single generator (Shirazi et al., 24 Aug 2025).

In discrete spaces the theory becomes especially rigid. If $A\in\mathcal I$4 is discrete with $A\in\mathcal I$5, then an action is $A\in\mathcal I$6-topologically stable if and only if it is an isolated point of $A\in\mathcal I$7 for the topology induced by $A\in\mathcal I$8. This gives a precise perturbative interpretation of stability in the totally disconnected setting (Shirazi et al., 24 Aug 2025).

4. Connectedness modulo an ideal and the compactification picture

A parallel topological development replaces ordinary connectedness by connectedness modulo an ideal $A\in\mathcal I$9 of subsets of a space $\mathcal I$00. A continuous mapping $\mathcal I$01 is 2-valued modulo $\mathcal I$02 if

$\mathcal I$03

and $\mathcal I$04 is connected modulo $\mathcal I$05 when no such continuous map exists. Equivalently, $\mathcal I$06 is connected modulo $\mathcal I$07 if and only if there is no separation for $\mathcal I$08 modulo $\mathcal I$09 (Koushesh, 2014).

For completely regular spaces, the theory is translated into the Stone–Čech compactification by the open subspace

$\mathcal I$10

The foundational equivalence is

$\mathcal I$11

This converts an ideal-relative topological property on $\mathcal I$12 into an ordinary connectedness statement on a compact remainder (Koushesh, 2014).

Two specialized forms are especially important. If $\mathcal I$13 is the ideal generated by open subsets of $\mathcal I$14 with pseudocompact closure, then

$\mathcal I$15

If $\mathcal I$16 is normal and $\mathcal I$17 is the ideal generated by closed realcompact subspaces, then

$\mathcal I$18

Here $\mathcal I$19 denotes the Hewitt realcompactification (Koushesh, 2014).

This framework also exhibits its own stability theory. Connectedness modulo an ideal is preserved under continuous surjections when the ideal is pulled back appropriately, is stable under finite unions under non-small intersection hypotheses, and may be transferred from a dense subspace under the condition $\mathcal I$20. By contrast, products are subtle: the paper formulates an explicit open question for products, and in the special pseudocompactness and realcompactness settings it supplies negative examples for naive product stability (Koushesh, 2014).

A plausible implication is that “topological stability modulo an ideal” should not be read exclusively as perturbation stability of dynamical systems. It also names a general topological method: weaken a classical property by allowing failure on subsets declared small by an ideal, and then analyze the resulting property through compactification or remainder spaces.

5. Graded and $\mathcal I$21-adic stability in augmented rings

In commutative algebra, Chang studies a commutative augmented ring $\mathcal I$22 whose augmentation ideal $\mathcal I$23 satisfies that $\mathcal I$24 is torsion of exponent $\mathcal I$25. The successive quotients

$\mathcal I$26

form the homogeneous pieces of the associated graded ring

$\mathcal I$27

The main theorem asserts the existence of $\mathcal I$28 such that

$\mathcal I$29

Equivalently, the isomorphism class of $\mathcal I$30 is eventually constant (Chang, 2019).

The proof passes through the graded $\mathcal I$31-algebra

$\mathcal I$32

which is generated in degree $\mathcal I$33 and is Noetherian, together with additive invariants

$\mathcal I$34

for finite abelian groups. A classification lemma then identifies the isomorphism class of a finite abelian group from the full collection of values $\mathcal I$35, and the graded-Noetherian argument shows these invariants eventually become constant in $\mathcal I$36 (Chang, 2019).

The resulting stability is explicitly noncanonical. The paper proves equality of isomorphism types of $\mathcal I$37 and $\mathcal I$38; it does not produce a distinguished identification between successive layers. This distinction matters conceptually: the filtration does not become constant, but its relative increments do (Chang, 2019).

The topological interpretation uses the $\mathcal I$39-adic topology, in which the powers $\mathcal I$40 form a neighborhood basis of $\mathcal I$41. Then the quotients $\mathcal I$42 describe the successive infinitesimal layers of the filtration. Chang’s theorem says that these layers eventually have constant structure as finite abelian groups, so the local incremental profile of the $\mathcal I$43-adic topology stabilizes. The inverse system

$\mathcal I$44

therefore has a tail in which each step is an extension by the same finite abelian group $\mathcal I$45. The paper also records the homological identification

$\mathcal I$46

so this stabilization transfers to a family of derived invariants (Chang, 2019).

This suggests an algebraic analogue of topological stability modulo an ideal: not perturbative stability of an action, but eventual regularity of the infinitesimal neighborhoods determined by an ideal-adic filtration.

6. Stabilization of prime spectra and homological invariants under powers of ideals

A second algebraic line studies stability phenomena for powers of ideals in graded rings. For squarefree principal Borel ideals $\mathcal I$47, the sets $\mathcal I$48 stabilize by Brodmann’s theorem, and the paper determines both persistence and the stable set of associated primes. A prime $\mathcal I$49 is persistent with respect to $\mathcal I$50 if

$\mathcal I$51

and squarefree strongly stable principal ideals satisfy the persistence property. The paper also computes the index of stability of the graded maximal ideal $\mathcal I$52, proving that $\mathcal I$53 for some, equivalently all large, $\mathcal I$54 if and only if

$\mathcal I$55

for the Borel generator $\mathcal I$56, and then gives the explicit formula

$\mathcal I$57

in terms of the interval-and-gap decomposition of the support of $\mathcal I$58 (Aslam, 2013).

For polymatroidal ideals, the stabilization problem is formulated through the indices

$\mathcal I$59

and

$\mathcal I$60

The paper proves $\mathcal I$61 in three cases: $\mathcal I$62 matroidal with $\mathcal I$63; $\mathcal I$64 polymatroidal with $\mathcal I$65 and $\mathcal I$66; and $\mathcal I$67 polymatroidal of degree $\mathcal I$68. It also gives a counterexample to the Herzog–Qureshi conjecture by exhibiting a polymatroidal ideal with $\mathcal I$69, and a family for which

$\mathcal I$70

Thus associated primes and depth need not stabilize simultaneously, even in highly structured monomial classes (Karimi et al., 2018).

These results sharpen an important conceptual point. Stability modulo powers of an ideal is often multi-layered: one invariant may stabilize immediately, while another continues to change. In the Borel and polymatroidal settings, the stabilized object may be a set of associated primes, a depth function, or a homological threshold rather than a dynamical action or a connectedness property. The shared theme is asymptotic regularity under iteration of an ideal operation.

Across these domains, topological stability modulo an ideal is therefore best understood as a family of ideal-relative stabilization principles rather than a single doctrine. In uniform dynamics it is a semiconjugacy-based perturbation property controlled by shadowing and expansivity (Shirazi et al., 24 Aug 2025). In general topology it appears as connectedness after ignoring $\mathcal I$71-small subsets and is encoded by the connectedness of a Stone–Čech remainder (Koushesh, 2014). In commutative algebra it describes stationary graded layers, persistent associated primes, or stabilized depth along ideal powers (Chang, 2019, Aslam, 2013, Karimi et al., 2018). What unifies these settings is the same structural move: an ideal marks the exceptional part, and stability is asserted only after those exceptions have been absorbed into the ideal.