Topological Stability of Random $cw$-Expansive Systems with Shadowing

Abstract: This paper investigates the topological stability of random dynamical systems . Our main goal is to extend the classical result of Walters \cite{Walters77} to the random setting by employing the notions of random continuum-wise expansivity and shadowing property. We prove that any random dynamical system that is random $cw$-expansive and satisfies the random shadowing property is randomly topologically stable. Furthermore, in the random setting we establish the topological invariance of these properties under conjugacy and analyze the relationship between topological transitivity and the periodic shadowing property.

• The paper introduces random cw-expansiveness and demonstrates that its combination with the shadowing property leads to random topological stability.
• It extends classical deterministic results by adapting continuum-wise concepts to stochastic dynamics through measurable bundle homeomorphisms.
• The work establishes key invariance and structural properties with practical implications for robust, noise-perturbed physical systems.

Topological Stability of Random $cw$-Expansive Systems with Shadowing

Overview

The paper "Topological Stability of Random $cw$-Expansive Systems with Shadowing" (2607.10505) rigorously extends the seminal results of Walters on the topological stability of deterministic expansive systems with shadowing to a broad class of random dynamical systems driven by external stochasticity. The main contributions are the formalization of random continuum-wise expansiveness ($cw$-expansiveness) and the shadowing property in the random setting, and the demonstration that their conjunction implies random topological stability. The work further elaborates several nontrivial invariance and structural properties, and clarifies the relationships between various notions of dynamical stability and recurrence in the random context.

Random Dynamical Systems and Key Definitions

Random dynamical systems (RDS) are modeled as skew product transformations $F(w, x) = (\theta(w), f_w(x))$ over a metric base $(\Omega, \mathcal{F}, \mathbb{P}, \theta)$, where $\theta$ is a $\mathbb{P}$-preserving invertible transformation and $\{f_w: U_w \to U_{\theta(w)}\}$ is a bundle of homeomorphisms. This formalism permits the systematic treatment of stochastic perturbations or noise-driven evolution. The notion of topological conjugacy is extended to the RDS framework via measurable bundle homeomorphisms preserving the cocycle structure.

The fundamental shift in this work lies in adopting $cw$-expansiveness, originally due to Kato, to random systems. In contrast to classical (pointwise) expansiveness, where all distinct points eventually separate uniformly, $cw$-expansiveness only requires that every nontrivial continuum eventually stretches beyond a random scale along typical random orbits. This non-pointwise notion is strictly weaker, enabling the treatment of dynamically rich systems beyond classical hyperbolicity.

Analogously, the shadowing property is generalized: random shadowing ensures that $cw$0-pseudo-orbits (i.e., sequences with small stepwise errors, defined over random trajectories) can be tracked within any prescribed accuracy $cw$1 by a true orbit for almost every $cw$2. The uniqueness of shadowing is nontrivial in the random setting due to the lack of uniformity, but the authors provide a precise statement under $cw$3-expansiveness.

Main Theoretical Results

Topological Invariance and Structural Properties

The paper establishes that both random $cw$4-expansiveness and the random shadowing property are invariant under random topological conjugacy. Thus, qualitative dynamical features are preserved under measurable coordinate changes, ensuring that the stability theory applies to entire conjugacy classes.

Furthermore, the authors show that random $cw$5-expansiveness and random equicontinuity are mutually exclusive on uncountable, locally connected spaces, echoing deterministic dichotomies between chaos and regularity. They also prove that $cw$6-expansiveness implies random sensitivity under mild assumptions, ensuring positive scale orbit divergence from arbitrary neighborhoods with positive probability.

Random Topological Stability

The central theorem is a random analog of Walters' result:

If a random dynamical system is random $cw$7-expansive and has the random shadowing property, then it is randomly topologically stable.

This means that for every random accuracy $cw$8, there exists a perturbation threshold $cw$9 such that any random homeomorphism $cw$0 with $cw$1 (almost surely) is semiconjugate back to $cw$2 by a family of random homeomorphisms $cw$3 with $cw$4. This formalizes stochastic robustness: qualitative dynamical features persist under small random perturbations.

Periodic and Finite Shadowing, and Structural Recurrence

The newly introduced periodic shadowing property for RDS, which requires the shadowing of periodic pseudo-orbits by genuine periodic orbits over periodic fibers, is established for random $cw$5-expansive systems with shadowing. The argument hinges on the uniqueness of shadowing and the structure of the base transformation. Further, it is shown that the finite shadowing property ensures the full (bi-infinite) shadowing property, mirroring deterministic results.

The periodic shadowing property also ensures the $cw$6-shadowing property (shadowing along the orbit of any periodic base point) in topologically mixing random systems, thereby linking strong mixing to orbit approximation and reinforcing the robustness of recurrence in the stochastic setting.

Numerical and Structural Implications

The theoretical results are concretized through examples including:

• Random homeomorphisms on the torus $cw$7 (involving irrational rotations and hyperbolic automorphisms) whose random compositions are both $cw$8-expansive and possess random shadowing, thus are randomly topologically stable.
• Random perturbations of Anosov diffeomorphisms, where standard hyperbolic random cocycle theory provides the necessary structural properties.

These examples illustrate that even under substantial stochastic variability, systems can exhibit robust and shadowable continuum-wise expanding dynamics.

Implications, Contrasts, and Future Prospects

The main result demonstrates that continuum-wise scenarios—significantly broader than pointwise or uniformly expansive regimes—retain strong topological rigidity under randomness, provided shadowing holds. This substantially enlarges the class of stochastically stable systems to include those not covered by classical hyperbolic theory.

The topological invariance of $cw$9-expansiveness and shadowing under random conjugacy is particularly important for classification problems in random dynamics. The methods developed set a new baseline for the analysis of random dynamical systems with weak expansiveness, suggesting that further weakening (e.g., entropy expansiveness, measure expansiveness) could be feasible future directions.

There are concrete practical implications for the structural stability of randomly perturbed physical systems, especially in cases where classical hyperbolic or deterministic methods are inapplicable. The results also connect to random symbolic dynamics, random hyperbolic attractors, and might inform the study of numerical discretizations for PDEs with stochastic components.

Open theoretical developments include the characterization of the stability boundary for random systems without the shadowing property, the extension to random noninvertible or partially defined systems, and exploration of analogous measure-theoretic rigidity phenomena for SRB measures or random equilibrium states.

Conclusion

This paper provides a rigorous and systematic extension of stability theory to random $F(w, x) = (\theta(w), f_w(x))$0-expansive systems with shadowing, proving that these systems are randomly topologically stable and establishing a suite of invariant and structural properties. This marks a significant expansion of the known classes of topologically robust random dynamical systems, linking continuum-wise phenomena, shadowing, and stochastic stability in a unified framework (2607.10505).