Syndetic Equicontinuity & Thick Sensitivity

• Syndetic equicontinuity and thick sensitivity are dynamical invariants defined via combinatorial return-time sets that distinguish tame recurrence from deep chaotic divergence.
• They refine classical notions of equicontinuity and sensitivity by establishing dichotomies in minimal and transitive systems using syndetic and thick structures.
• These invariants extend to non-autonomous dynamics, offering a framework to analyze order–chaos transitions through factor maps and maximal equicontinuous factors.

Syndetic equicontinuity and thick sensitivity are advanced dynamical invariants situated between classical equicontinuity and strong chaos, formulated via return-time sets characterized by combinatorial largeness (syndeticity, thickness) rather than mere nonemptiness. These notions have become central both in general topological dynamics and in extensions to non-autonomous systems, yielding refined dichotomy theorems generalizing the classical Auslander–Yorke dichotomy for minimal and transitive systems.

1. Foundational Concepts: Furstenberg Families, Syndeticity, and Thickness

A Furstenberg family $\mathscr{F}$ is a hereditary upward family of subsets of $\mathbb{N}$: if $F_1 \in \mathscr{F}$ and $F_1 \subset F_2$, then $F_2 \in \mathscr{F}$. The dual family $k\mathscr{F}$ consists of sets in $\mathbb{N}$ that intersect every $F \in \mathscr{F}$. A family is translation-invariant if, whenever $F \in \mathscr{F}$ and $m \ge 0$, both $F + m$ and $(F - m) \cap \mathbb{N}$ belong to $\mathscr{F}$.

Two combinatorially important families are:

• Thick sets: $F \subset \mathbb{N}$ is thick if for every $L$, there exists an interval of length $L$ in $F$.
• Syndetic sets: $S \subset \mathbb{N}$ is syndetic if there exists $N$ so that every block of $N$ consecutive integers meets $S$.

Duality holds: $k\mathscr{F}_{\rm thick} = \mathscr{F}_{\rm syn}$, and $k\mathscr{F}_{\rm syn} = \mathscr{F}_{\rm thick}$ (Ju et al., 2019).

2. Definitions: F-equicontinuity, Syndetic Equicontinuity, and Thick Sensitivity

Given a compact metric space $(X,d)$ and $T: X \to X$ continuous, let $\mathscr{F}$ be a Furstenberg family.

• $\mathscr{F}$-equicontinuity: $(X,T)$ is $\mathscr{F}$-equicontinuous if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $d(x,y) < \delta$ implies $$\{n \in \mathbb{N} : d(T^n x, T^n y) < \varepsilon\} \in \mathscr{F}$$. A point $x$ is $\mathscr{F}$-equicontinuous if this holds for all $y$ sufficiently close to $x$.
  • Syndetic equicontinuity: Take $\mathscr{F} = \mathscr{F}_{\rm syn}$. For every $\varepsilon>0$, there exists $\delta>0$ such that $d(x,y)<\delta$ forces

  $$\{n : d(T^n x, T^n y) < \varepsilon\} \ \text{is syndetic}.$$

• $\mathscr{F}$-sensitivity: There exists $\varepsilon>0$ such that for every nonempty open $U \subset X$,

$$\{n\in\mathbb{N}: \operatorname{diam}(T^n(U)) > \varepsilon \} \in \mathscr{F}.$$

• Thick sensitivity: $\mathscr{F} = \mathscr{F}_{\rm thick}$. For every open $U$, the set of times when images of $U$ have diameter $> \varepsilon$ is thick.

Alternative (pairwise) forms define syndetic equicontinuity via pairs $y, z$ in a neighborhood $U$ around $x$, requiring $\{n: d(T^n y, T^n z) < \varepsilon\}$ syndetic (Li et al., 2021).

3. Auslander–Yorke-Type Dichotomies and Their Refinements

Key Theorem: Given a Furstenberg family $\mathscr{F}$ whose dual $k\mathscr{F}$ is translation-invariant, any transitive system $(X,T)$ satisfies exactly one:

1. $(X,T)$ is $\mathscr{F}$-sensitive;
2. $(X,T)$ is almost $k\mathscr{F}$-equicontinuous (admits a $k\mathscr{F}$-equicontinuous point; for transitive points, this is residual) (Ju et al., 2019).

Specialized dichotomy: For $\mathscr{F} = \mathscr{F}_{\rm thick}$ (thick sensitivity) and dual $k\mathscr{F} = \mathscr{F}_{\rm syn}$ (syndetic equicontinuity):

• A transitive system is either thickly sensitive or almost syndetically equicontinuous.

Minimal system variant: If $(X,T)$ is minimal, the dichotomy strengthens: it is either thickly sensitive or (fully) syndetically equicontinuous (Li et al., 2021, Huang et al., 2015).

This refinement bridges the classical alternative (“minimal equicontinuous vs. sensitive system”) by weakening equicontinuity (to syndetic equicontinuity) and strengthening sensitivity (to thick sensitivity).

4. Relationships and Structural Properties

The interplay between syndetic equicontinuity and thick sensitivity reflects combinatorial properties of return-time sets:

• If a system is thickly sensitive, no point can have the strong “syndetic $\varepsilon$-recurrence” property, so exactly one persists in the dichotomy.
• In minimal systems, thick sensitivity, multi-sensitivity, and thickly syndetical sensitivity are equivalent, each characterized by arbitrarily long runs (“thickness”) in the set of large-diameter times (Huang et al., 2015).
• If a factor map onto a maximal equicontinuous factor is almost one-to-one, thick sensitivity fails; such systems are syndetically equicontinuous (Huang et al., 2015, Huang et al., 2015).
• For non-autonomous periodic systems in uniform spaces or $T_3$ topological spaces, analogues of the dichotomy persist, with thick sensitivity and syndetic topological equicontinuity replacing their classical forms (Malik et al., 27 Dec 2025).

These properties are preserved under open factor maps, and syndetic equicontinuity is inherited by factors.

5. Examples and Illustrative Classes

System Class	Thick Sensitivity	Syndetic Equicontinuity	Reference
Circle rotation	No	Yes	(Li et al., 2021)
Full one-sided (or two-sided) shift	Yes (all $n$)	No	(Li et al., 2021)
Sturmian subshift	No	Yes	(Huang et al., 2015)
Minimal almost 1-1 extension	No	Yes	(Huang et al., 2015)
Weakly mixing system with $n$ minimal points	Yes (for $n$)	No	(Li et al., 2021)

• Rotations and most almost automorphic systems: syndetically equicontinuous, not thickly sensitive.
• Full shift: mixing and thickly sensitive for all $n$.
• Sturmian minimal subshifts: syndetically equicontinuous but fail classical equicontinuity.
• Weakly mixing systems with finitely many minimal points: thickly sensitive according to the number of minimal orbits.

6. Extensions: Non-Autonomous Settings and Generalizations

For periodic non-autonomous dynamical systems on uniform or topological spaces, the dichotomous framework persists (Malik et al., 27 Dec 2025):

• Thick sensitivity is defined relative to arbitrary periodic iteration sequences.
• Syndetic equicontinuity requires uniform closeness along syndetic sets of times, adapted to non-autonomous context.
• For minimal periodic non-autonomous systems, exactly one of thick sensitivity or syndetic equicontinuity (topological analogues exist: Hausdorff sensitivity vs. topological equicontinuity) prevails.

Multi-sensitivity is equivalent to thick sensitivity in minimal and transitive settings, and similar factor map criteria determine the failure of (thick) sensitivity.

7. Relevance and Structural Implications

Syndetic equicontinuity provides a robust invariant for distinguishing “tame” recurrent behaviors from expansive chaos, sensitive not merely to possible divergence but to the combinatorial structure of times at which orbits remain close. Thick sensitivity encapsulates the presence of chaotic behavior in a strong, block-recurrent sense, where separated orbits can be witnessed in arbitrarily long consecutive intervals.

The dichotomies and associated structure theorems have wide applicability:

• They refine the chaos/equicontinuity alternative in minimal and transitive systems.
• They enable construction and classification of systems (e.g., via maximal distal or almost one-to-one factors).
• They extend naturally to higher-order ($n$-tuplewise) settings and to non-autonomous dynamics, emphasizing the universality of these combinatorial dynamical dichotomies (Liu et al., 2019, Huang et al., 2015, Malik et al., 27 Dec 2025).

These invariants also bridge to mean equicontinuity and measure-theoretic chaos, and provide a framework for further combinatorial generalizations beyond syndetic or thick families (Ju et al., 2019).