CW2-Expansive Homeomorphisms in Surface Dynamics

• CW₂-expansive homeomorphisms are surface mappings that weaken classical expansivity by controlling the behavior of connected sets while permitting limited nontrivial orbit fibers.
• They exhibit key structural features such as isolated spines, local arc structures, and finite spectral decompositions that bridge the gap between Anosov dynamics and general homeomorphisms.
• Recent constructions and generic approximation methods reveal practical insights into dynamical rigidity, entropy behavior, and the nature of anomalous stable sets.

A continuum-wise 2-expansive (cw₂-expansive) homeomorphism is a specific weakening of classical expansivity, adapted to the behavior of nontrivial connected sets under iteration by surface homeomorphisms. In this context, the dynamics restrict the size and nature of orbit fibers for connected sets but allow considerably more flexibility than pointwise (classical) expansive dynamics. This concept is central to the study of intermediate dynamic rigidity between Anosov or pseudo-Anosov maps and general homeomorphisms and connects to the structure of surface foliations, hyperbolic dynamics, and topological entropy.

1. Formal Framework and Definitions

Let $(X, d)$ be a compact metric space and $f: X \to X$ a homeomorphism.

• Continuum: A non-empty compact connected subset $C \subset X$.
• cw-expansive: $f$ is continuum-wise expansive (cw-expansive) if there exists $c > 0$ such that whenever $C \subset X$ is a continuum with $\operatorname{diam}(f^n(C)) \leq c$ for all $n \in \mathbb{Z}$, then $C$ is a singleton.
• cw$_N$-expansive: There exists $c > 0$ such that for every $x \in X$, $\#(C^s_c(x) \cap C^u_c(x)) \leq N$, where $C^s_c(x)$ (resp. $C^u_c(x)$) is the connected component of the $c$-stable (resp. $c$-unstable) set containing $x$.
• cw₂-expansive: The special case $N = 2$, i.e., $\#(C^s_c(x) \cap C^u_c(x)) \leq 2$ for every $x$.
• cw-local product structure: For every $\varepsilon > 0$, there exists $\delta > 0$ such that if $d(x, y) < \delta$, then $$C^s_\varepsilon(x) \cap C^u_\varepsilon(y) \neq \emptyset$$.

cw₂-expansivity sits in a hierarchy: $$\text{expansive} \implies 2\text{-expansive} \implies \text{cw}_2\text{-expansive} \implies \text{cw-expansive}$$ Each implication is strict on compact surfaces (Sarmiento et al., 31 Dec 2025, Artigue et al., 2020).

2. Structural Properties and Classification

Classical expansive homeomorphisms on surfaces are conjugate to Anosov or pseudo-Anosov diffeomorphisms, which exhibit locally connected foliations and strong hyperbolic features. cw₂-expansive homeomorphisms, though strictly more general, retain many structural features:

• Isolation of Spines: The sets where stable and unstable continua ("spines") meet are isolated and can be classified via bi-asymptotic sectors—closed disks bounded by stable and unstable arcs (Arruda et al., 2023). Regular bi-asymptotic sectors contain exactly one spine; every spine lies in a small sector neighborhood free of other spines. Thus, the collection of spines is at most countable and typically finite for "hyperbolic" cases (cw$_3$ implies finite spines, hence cw₂) (Arruda et al., 2023).
• Local Arc Structure: For a cw$_F$-hyperbolic homeomorphism, local stable and unstable continua are arcs—either $[0,1]$ at a spine or $[-1,1]$ in regular points. This follows from adaptations of the techniques of Hiraide and Lewowicz for classical expansive dynamics.
• Examples Distinguishing Classes: On the sphere $S^2$, the quotient of a linear Anosov on $\mathbb{T}^2$ by the antipodal involution produces a pseudo-Anosov which is cw₂-hyperbolic but not cw₁-expansive; the same holds for the $n$-fold product results on more general surfaces (Arruda et al., 2023, Artigue et al., 2020, Sarmiento et al., 31 Dec 2025).
• Spectral Decomposition: cw₂-hyperbolic dynamics admit a finite spectral decomposition into finitely many chain-recurrent classes, as in classical hyperbolic theory (Artigue et al., 2020).

3. Genericity and Density

A fundamental property of cw-expansivity (and, by implication, cw₂-expansivity) on closed surfaces is its genericity:

• Density: For any homeomorphism $f$ of a closed surface $M$ and any $\varepsilon > 0$, there exists a cw-expansive homeomorphism $g$ with $d_{C^0}(f, g) < \varepsilon$ (Artigue, 31 Mar 2025).
• Generic Almost cw-expansivity: For a generic homeomorphism $f$ (residual set in the $C^0$-topology), for every $\varepsilon > 0$, there exists a cw-expansive $g$ arbitrarily $C^0$-close and a continuous, monotone semiconjugacy $\pi: M \to M$ with fibers of diameter $< \varepsilon$ that do not separate $M$ (Artigue, 31 Mar 2025). This indicates that typical surface dynamics are, up to small monotone extension, closely approximated by continuum-wise expansive behavior.

4. Explicit Constructions and Anomalous Phenomena

Surface homeomorphisms that are cw₂-expansive but not $N$-expansive for any finite $N$ (and not classically expansive) have been rigorously constructed:

• Connected-sum Constructions: By gluing DA-Anosov type dynamics with stable and unstable plugs, one obtains 2-expansive (hence cw₂-expansive) but not expansive examples on any genus $g \geq 2$ surface (Sarmiento et al., 31 Dec 2025).
• Sphere/Torus Examples Not $N$-Expansive: On both sphere and torus, there are cw₂-expansive homeomorphisms with arbitrarily large dynamic balls, built using pseudo-Anosov with 1-prong singularities and DA-plugs (Sarmiento et al., 31 Dec 2025).
• Anomalous Stable Sets: Plug constructions allow for fixed points with non-locally connected, yet connected, local stable sets—this persists in cw₂-expansivity (Artigue, 2015, Sarmiento et al., 31 Dec 2025). The canonical example is the insertion of a "topologist's comb" (non-locally connected continuum) as a local stable set at a saddle fixed point, carried through suitable DA-gluing surgery.

A summary table of key constructions:

Surface	Construction Type	Notion Satisfied
Genus $g \geq 2$	DA-gluing, connected sum	2-expansive, cw₂-expansive
$\mathbb{S}^2$	Pseudo-Anosov + DA-plug	cw₂-expansive, not $N$-expansive
$\mathbb{T}^2$	Anomalous plug	cw₂-expansive, anomalous stable set

5. Dynamical and Topological Consequences

cw₂-expansive (hyperbolic) dynamics enforce significant constraints and enable the derivation of important properties:

• L-Shadowing Property: cw₂-hyperbolic systems (those with a local product structure) have the L-shadowing property: for sequences with vanishing pseudo-orbit error at infinity, there are points that both $\varepsilon$-trace and asymptotically shadow them, generalizing classical shadowing (Artigue et al., 2020).
• Finiteness of Periodic Points: In the hyperbolic case, for each $k$, there are only finitely many period-$k$ points (Artigue et al., 2020).
• Non-Entropy-Expansive: Any genuinely non-expansive cw₂-hyperbolic system contains arbitrarily small disconnected semihorseshoes, so it is not entropy-expansive (Artigue et al., 2020).
• Monotone Extensions and Quotients: If $f$ is a small $C^0$-perturbation of a cw-expansive homeomorphism $g$, the relation of lying in a small-diameter stable continuum gives a monotone equivalence relation whose quotient inherits the cw-expansive property (Achigar et al., 2017, Artigue, 31 Mar 2025).

6. Hierarchy, Rigidity, and Contrasts

cw₂-expansivity captures a class strictly larger than Anosov/pseudo-Anosov dynamics, yet still robust enough to exclude pathological behaviors ubiquitous in general homeomorphisms:

• Expansivity Chain: On compact surfaces, each inclusion in

$$\{\text{expansive}\} \subsetneq \{2\text{-expansive}\} \subsetneq \{\text{cw}_2\text{-expansive}\} \subsetneq \{\text{cw-expansive}\}$$

is proper, witnessed by explicit construction (Sarmiento et al., 31 Dec 2025).

• Rigidity Under Shadowing: On surfaces, the combination "shadowing + countably- or entropy cw-expansive" is rigid: the only possibility is conjugacy to an Anosov diffeomorphism, i.e., truly expansive (Artigue et al., 2019).
• Anomalous Sets: Stable sets need not be locally connected in cw₂-expansive systems; products and plug insertions generate Cantor-type or non-locally connected stable continua—impossible in classical expansive theory (Artigue, 2015, Sarmiento et al., 31 Dec 2025).
• Generic Behavior: Generic (in the sense of Baire category) surface homeomorphisms are almost cw-expansive through semiconjugacy to nearby cw-expansive maps, with fibers that are small non-separating continua (Artigue, 31 Mar 2025).

7. Outlook and Open Directions

Current research elucidates the range of behaviors allowed by cw₂-expansivity, but several questions remain:

• The interaction between cw₂-expansivity and other dynamical properties (e.g., measure-theoretic mixing, entropy structure) is not exhausted.
• The topological types of non-locally connected stable sets in low-genus surfaces are not fully classified.
• Classification up to conjugacy and the existence of normal forms for cw₂-hyperbolic dynamics beyond classical Anosov/pseudo-Anosov maps are open for investigation.

Recent works by Artigue, Carvalho, Cordeiro, Vieitez, Arruda, Sarmiento, and others (Artigue et al., 2020, Sarmiento et al., 31 Dec 2025, Arruda et al., 2023, Artigue, 31 Mar 2025, Artigue, 2015, Achigar et al., 2017) provide a robust foundation for further exploration of continuum-wise expansive surface dynamics, offering both general theory and pathological counterexamples.