Shadowing, transitivity and a variation of omega-chaos

Abstract: We study a special type of shadowing (DSP) of chain transitive continuous self-maps of compact Hausdorff spaces. We prove some basic properties of DSP. As application of DSP, we obtain sufficient conditions for a statistical variant of $ω$-chaos and prove the topological genericity of it. We also consider topological distribution of irregular points under the assumption of DSP.

• The paper introduces the D-shadowing property (DSP) that enhances classical shadowing through cyclic decompositions and chain transitivity.
• It demonstrates that DSP in compact systems with positive entropy leads to strong distributional chaos, including a statistical variant of omega-chaos.
• Utilizing techniques like Vietoris topology and inverse limit models, the study provides structural insights into chaotic dynamics and irregular behavior.

Shadowing, Transitivity, and a Variation of Omega-Chaos

Introduction and Context

This paper introduces and investigates a refined shadowing property (the D-shadowing property, abbreviated DSP) for chain transitive continuous self-maps on compact Hausdorff spaces. The study situates itself at the intersection of classical shadowing in dynamical systems, the topological theory of chaos, and entropy-driven phenomena, and relates DSP to statistical and topological variants of chaos, notably a statistical refinement of omega-chaos ($\bar{\omega}$-chaos). The work addresses open questions regarding the sufficient conditions for strong forms of distributional chaos under topological entropy and shadowing assumptions, generalizing and extending previous partial results on zero-dimensional systems.

Core Concepts: D-Shadowing Property (DSP)

The DSP builds on the classical shadowing property—where pseudo-orbits can be traced arbitrarily closely by true orbits—by incorporating an equivalence relation based on chain transitivity and specific partitions (clopen decompositions) of the phase space. The main technical construction involves equivalence relations $\sim_{f,\mathcal{U}}$ defined via the structure of U-chains and their cycles for finite open covers $\mathcal{U}$, ultimately leading to a cyclic decomposition of the space and the identification of chain proximal pairs.

The DSP is then defined through three equivalent conditions, which combine shadowing across all classes of the decomposition and continuity properties on the space of closed subsets (with the Vietoris topology). Notably, the paper establishes that DSP is strictly stronger than the classical shadowing property, inheriting but not limited to its consequences.

Main Theoretical Results

Fundamental Properties and Structure

Equivalence of DSP Definitions: The paper rigorously proves the equivalence of several formulations of DSP for chain transitive continuous self-maps, involving the covering and shadowing of all classes in the cyclic decomposition. This establishes that chain transitive systems with DSP also have the usual shadowing property, but the converse is nontrivial, especially for higher-dimensional or non-metrizable compact spaces.

Continuity and Factorization: DSP is shown to be preserved under taking factors, provided the factor and the original system satisfy natural compatibility and shadowing properties. Additionally, it is demonstrated that in zero-dimensional spaces with chain transitive and shadowing maps, DSP necessarily holds, building on recent structural results about the inverse limits of subshifts of finite type.

Relationship to Chaos and Entropy

Implication for Distributional Chaos: The presence of DSP, combined with chain transitivity and positive topological entropy, is shown sufficient for the system to exhibit strong forms of distributional chaos—most notably, in the sense of DC1. This significantly advances prior results, confirming conjectures in the zero-dimensional case and extending applicability to much broader settings. The result is leveraged via factor and extension arguments, the use of Mycielski sets, and the construction of scrambled sets within each class of the cyclic partition.

Statistical Omega-Chaos: The paper introduces a statistical variant of omega-chaos ($\bar{\omega}$-chaos), where the existence of uncountable scrambled sets is defined using densities of visits to neighborhoods and set-theoretic properties of omega-limit sets. It is proven that DSP, chain transitivity, and positive entropy guarantee the existence of $\bar{\omega}$-scrambled, continuum-dense Mycielski sets in every cylinder class, with the further property that for interval maps, $\bar{\omega}$-chaos is equivalent to positive topological entropy—a result that does not generalize to all compact spaces.

Irregular Points and Historical Behavior

A significant advancement is made in understanding the topological distribution of irregular points (those with non-convergent Birkhoff averages, i.e., historic behavior). The paper shows that under DSP (and chain transitivity), for every continuous observable, the set of irregular points is either empty or residual in every component of the cyclic decomposition. Under positive entropy, one can always find an observable with a non-empty, residual irregular set.

Methodology and Technical Innovations

• Cyclic Decompositions and Equivalence Relations: The construction and analysis of open, closed, and $\sim_{f,\mathcal{U}}$-invariant equivalence classes underpin all results, providing a canonical partition of the phase space compatible with the dynamics.
• Vietoris Topology and Continuity Criterion: The paper leverages continuity of the class map in the Vietoris topology to establish topological genericity of dynamical features.
• Inverse Limits and Zero-Dimensional Models: By relating general systems with DSP to inverse limits of subshifts of finite type (via the Mittag-Leffler condition), the study grounds the structure theory in the best-understood symbolic settings.
• Statistical Methods for Scrambled Sets: Mycielski’s theorem and density arguments are systematically applied to construct large scrambled sets with prescribed omega-limit properties.

Numerical and Separation Results

• Equivalence with Entropy on the Interval: The equivalence between $\bar{\omega}$-chaos and positive topological entropy for interval maps is established, providing a sharp separation from higher-dimensional or more general compact spaces, where such equivalence fails.
• Residuality of Irregular Sets: The construction yields residual scrambled sets and residual sets of irregular points within each cyclic class, confirming robustness and topological largeness of chaotic regimes.
• Factor/map-based Transfer: The theorems include explicit factor maps and their impact on dynamical properties, showing how DSP and chaos can be inherited across extensions, especially for totally disconnected factors.

Implications and Future Directions

The theoretical implications are multifold:

• Classification of Dynamical Complexity: The findings provide new invariants and dichotomies for classifying the complexity of chain transitive dynamical systems on compact spaces, especially in relation to shadowing variations and the density of chaotic and historic behavior.
• Limitations of Shadowing: While DSP implies many strong forms of chaos, the absence of equivalence with classical omega-chaos outside the metrizable or zero-dimensional setting points toward structural limitations and nuanced distinctions in the landscape of chaos.
• Genericity in Manifolds: For generic dynamics (in the Baire sense) on closed differentiable manifolds and in the space of homeomorphisms, $\bar{\omega}$-chaos is shown to be prevalent, suggesting that DSP-like properties are generic in broad classes of dynamics.

Future developments may focus on:

• Determining precisely for which classes of compact spaces DSP and the classical shadowing property coincide.
• Extending the equivalence between positive entropy and $\bar{\omega}$-chaos to larger classes of spaces or weakening the topological requirements.
• Exploring finer stratifications of irregular sets beyond residuality, possibly refining entropy and measure-theoretic classification.

Conclusion

This work systematically develops the D-shadowing property, thoroughly analyzes its structural, topological, and dynamical consequences, and establishes strong connections to entropy, distributional chaos, and statistical omega-chaos. The results resolve longstanding questions regarding the sufficiency of shadowing plus entropy for chaos in a wide class of systems, reinforce the prevalence of irregular and chaotic regimes, and provide a rigorous bridge between symbolic dynamics and general compact dynamical systems. The methods and constructions introduced set a new standard for the study of shadowing and chaos theory in topological dynamics.