Extensibility and denseness of periodic semigroup actions

TL;DR

This work extends the study of periodicity and invariant measures from groups to semigroups by formulating topological and measure-theoretic notions of periodicity and extensibility. Using natural extensions to the free $S$-group $\bm{\Gamma}$, it links the density of periodic (or ergodic periodic) measures to the extensibility of $S$-invariant measures, enabling transfer of the pa/epa properties from the group setting to semigroups. It proves that left amenable semigroups that are residually finite groups have the epa property, and that free semigroups arising from subsemigroups of free groups possess the pa property via $\text{Ext}_{\mathbf{F}_d}(\mathcal{A}^S,S)=\mathcal{M}_S(\mathcal{A}^S)$. A key finding is a measure-theoretic characterization: every $S$-invariant measure is $\mathbf{F}_d$-extensible if and only if the acting semigroup coincides with the free case, while non-free realizations admit fully supported Markov measures that fail to be extensible. Overall, the results provide concrete criteria for when finitely supported invariant measures are dense among all invariant measures for semigroup actions, with implications for symbolic dynamics and dynamical systems with non-invertible time evolution.

Abstract

We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup.

Figures (3)

• Figure 1: A diagram of the $\mathbb{F}_2^{+}$-orbit $\{x,y\}$ of $x$ and the representation of $x$ as a labeling of $\mathbb{F}_2^+$.
• Figure 2: The Cayley graph of $\mathbb{F}_2$ for the definition of a Markov measure on $\mathcal{A}^{\mathbb{F}_2}$. Dashed lines represent inverses of the generators of $\mathbb{F}_2$.
• Figure 3: In this cycle in $G$, transitions corresponding to solid edges all have high probability. As $A_{i_1}^{\varepsilon_1}\cdots A_{i_n}^{\varepsilon_n}\mathbf{v^*}\ne\mathbf{v^*}$, the final transition is thus forced to be of very low probability.

Theorems & Definitions (96)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 1.1
• Definition 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5