Overview
This study investigates entropy realization within continuous actions of infinite countable amenable groups. It introduces a specific characteristic termed the approximate product property for these group actions. This property is defined by its allowance for a small proportion of tracing mistakes on prescribed pairwise disjoint, sufficiently invariant finite sets. The research demonstrates that this approximate product property has implications for the entropy-related characteristics of invariant measures.
Research Context
The work is situated within the study of ergodic measures and their entropy properties in the context of continuous actions of infinite countable amenable groups. A central theme is the realization of entropy values, particularly understanding the range and distribution of entropies attainable by ergodic measures.
Approach
The research defines and utilizes an approximate product property specific to amenable group actions. This property allows for a limited number of tracing errors on predetermined, pairwise disjoint, and sufficiently invariant finite sets. The construction methodology integrates zero-entropy exact tilings with an estimation technique for the complexity of tracing mistakes, specifically a finite-block estimate.
Findings
- The introduced approximate product property implies both entropy-denseness and almost entropy-approximability for every invariant measure.
- The construction method combines zero-entropy exact tilings with a finite-block estimate for the complexity associated with tracing mistakes.
- Under conditions of asymptotic entropy expansiveness, the property of almost entropy-approximability is shown to upgrade to full entropy-approximability.
- For any value $\alpha$ within the range $0 \leq \alpha \leq h(X,G)$, where $h(X,G)$ represents the maximum entropy, ergodic measures with entropy $\alpha$ form a residual subset of invariant measures whose entropy is at least $\alpha$.
- A direct consequence of these findings is that the set of entropies for ergodic measures spans the entire interval $[0, h(X,G)]$.
Why This Matters
The findings clarify the range of possible entropy values for ergodic measures in continuous actions of infinite countable amenable groups, demonstrating that the set of ergodic measure entropies is continuous over a specific range. This provides a structural understanding of entropy realization in these dynamical systems.