Entropy structures with continuous partitions of unity

Abstract: Using only continuous partitions of unity, we provide equivalent definitions for the metric, topological and topological tail entropies and pressures of a continuous self-map of a compact set, as well as their conditional versions. A tail variational principle for these new definitions is proved. We extend Downarowicz's notions of candidates and entropy structures to account for almost-increasing sequences of functions arising from the new definitions. Finally, we deduce a partial answer to a question raised by Newhouse.

• The paper introduces continuous partitions of unity to rigorously redefine metric and topological entropy, proving equivalence with classical measures.
• It establishes a novel formulation for topological tail entropy and pressure, quantifying convergence gaps via vanishing-diameter sequences.
• The work extends entropy structures to weak-candidate frameworks, enabling precise computational approximations and broader applicability in non-symbolic settings.

Equivalent Entropy and Pressure Structures via Continuous Partitions of Unity

Introduction and Context

The paper "Entropy structures with continuous partitions of unity" (2603.29720) extends the framework of entropy and pressure in topological dynamical systems by developing definitions and structural results utilizing continuous partitions of unity. Traditionally, entropy (both metric and topological) as well as pressure notions are defined either via measurable or finite partitions, separated/spanning sets, or open covers. This work systematically formulates these invariants with continuous partitions of unity, providing alternative but equivalent formulations, and relates them to the entropy structures as formalized by Downarowicz.

The approach overcomes known obstructions in the literature concerning the use of partitions of unity for entropy structures—particularly highlighting subtle failures for certain refining sequences while providing positive results under natural relaxations.

Main Contributions

1. Entropy and Pressure via Continuous Partitions of Unity

The paper rigorously defines metric and topological entropy and their pressure counterparts using only continuous partitions of unity, generalizing the metric-entropy formalism of Ghys–Langevin–Walczak. The key findings include:

• Metric Entropy: The function $\mu \mapsto h_\mu(T)$, defined as the supremum of entropies over all continuous partitions of unity, coincides with the classical Kolmogorov–Sinai metric entropy. Analogous statements hold for pressure.
• Topological Entropy: The supremum over continuous partitions of unity also yields the classical topological entropy.
• Conditional Versions: Conditional entropies and pressures are formulated in this context, with precise and constructive definitions.

These results include formal subadditivity, existence of limits, and upper semi-continuity properties for the resulting entropy functions (Proposition $2603.29720$, Theorem 1).

2. Tail Entropy and Variational Principles

A new formulation for topological tail entropy is developed via continuous partitions of unity. The paper extends the tail variational principle to this setting:

• For any vanishing diameter sequence of continuous partitions of unity $(\Psi_k)$, the sequence of metric entropies $h_\mu(T,\Psi_k)$ converges pointwise to $h_\mu(T)$ for all invariant $\mu$.
• Uniform non-convergence is quantitatively described by tail entropy:

$$\lim_{k\to\infty} \sup_{\mu\in M(X,T)} h_\mu(T) - h_\mu(T,\Psi_k) = h^*(T)$$

where $h^*(T)$ is the (Misiurewicz) topological tail entropy.

Analogous principles and formulations are proved for topological tail pressure, fully mirroring metric-to-topological relationships in this context.

3. Entropy Structures and Weak-Candidate Extensions

The work relates the aforementioned constructions to entropy structures as developed by Downarowicz:

• The classical entropy structure is characterized by equivalence classes (under uniform equivalence) of non-decreasing sequences of upper semi-continuous functions converging to the entropy map.
• The paper shows that any sequence of continuous partitions of unity with vanishing diameter gives rise to a sequence of entropy functions which, with a mild relaxation (almost-increasingness), still belong to the entropy structure equivalence class after an extension of the equivalence notion to 'weak candidates' (Theorem 3).
  • This includes an extension whereby almost-increasing sequences are considered suitable, reflecting the non-strict monotonicity of entropy along general partition refinement sequences.

As a corollary, structural invariants including the symbolic extension entropy function, transfinite sequences, and super-envelopes are recovered from these weak-candidates.

4. Technical Properties and Implications

The approaches provide strong-quantitative versions of convergence (almost-uniform) and demonstrate that all known entropy invariants and order-of-accumulation results are stable under these new formulations (Proposition: transfinite sequences and superenvelopes are invariant within the weak class).

The equivalence between the “tilde” entropies defined via continuous partitions of unity and the classical definitions is established with constructive proofs, clarifying open issues in the literature regarding the sufficiency of such formulations.

Theoretical and Practical Implications

Theoretical Significance

• These results close subtle gaps in the theory of entropy structures for continuous maps on compact metric spaces, including non-invertible systems.
• By extending the equivalence of entropy structures to weak-candidates, the study broadens the class of practical entropy approximations without sacrificing invariance, which has ramifications for symbolic extension theory and the transfinite machinery developed for entropy structure depth.
• The direct formulations in terms of continuous functions greatly facilitate the analysis of entropy in non-symbolic settings and make available the rich machinery of continuous function spaces, enabling improved upper semi-continuity properties and various approximation arguments.

Practical and Computational Directions

• For computational dynamics and the construction of entropy-increasing approximations, the use of continuous partitions of unity admits practical advantages, allowing entropy and pressure to be evaluated via smooth approximations to characteristic functions.
• The vanishing-diameter condition offers an explicit and easily-checkable criterion for entropy cumulation, facilitating entropy estimation in practice.
• The extension of pressure and entropy tail principles directly informs the analysis of systems with high complexity or little structure (e.g., non-expansive maps, C$^0$-dynamics).

Impact and Future Directions

The structural extension of entropy theory to continuous partitions of unity in such generality points towards further applications:

• Extensions to Markov Operators: The extension of these notions to operators, particularly Markov operators as in [DF05], remains an open direction, with the current framework potentially providing the required technical tools.
• Refinements in Symbolic Extension Theory: The reinforced invariance under weak-candidates hints at the possibility of new universal formulas for symbolic extension entropy and associated invariants.
• Local Variational Principles: Open questions remain regarding fully local versions of the variational principle for arbitrary continuous partitions of unity.

Additionally, the approach opens a path to treat entropy and pressure in broader settings (e.g., non-compact or non-metrizable dynamical systems), given the robustness of the structural approach.

Conclusion

This paper provides a comprehensive and technically rigorous development of entropy and pressure invariants for continuous dynamical systems, using only continuous partitions of unity. The main results establish full equivalence with classical definitions, extend the concept of entropy structure to weak-candidates, and characterize tail entropy and pressure in this setting. These contributions fill critical gaps in the literature, clarify subtle foundational issues, and set the stage for further advances both in ergodic theory and dynamical systems on general topological spaces.