Measures of maximal entropy for suspension flows over the full shift

Abstract

We consider suspension flows with continuous roof function over the full shift Σ on a finite alphabet. For any positive entropy subshift of finite type Y ⊂ Σ, we explictly construct a roof function such that the measure(s) of maximal entropy for the suspension flow over Σ are exactly the lifts of the measure(s) of maximal entropy for Y . In the case when Y is transitive, this gives a unique measure of maximal entropy for the flow which is not fully supported. If Y has more than one transitive component, all with the same entropy, this gives explicit examples of suspension flows over the full shift with multiple measures of maximal entropy. This contrasts with the case of a Hölder continuous roof function where it is well known the measure of maximal entropy is unique and fully supported.

Measures of maximal entropy on subsystems of topological suspension semiflows

“…We discuss previous results in this direction and our approach. We extensively generalize our previous work [12] in which we proved that for the full shift (Σ, σ) and any positive entropy subshift of finite type Y ⊂ Σ, there exists a roof function such that the MMEs for the suspension flow over the full shift are exactly the lifts of the MMEs for the subshift. In the current paper, we remove the restriction that the subset Y is a shift of finite type, and we remove the need for the ambient space to be symbolic, allowing any topological dynamical system with upper semicontinuous entropy map.…”

“…We now turn to the phenomenon of non-uniqueness of MME. Examples of suspension flows over the full shift with non-unique MME were given in [12], however each MME was supported on a different transitive component. Theorem 3.1 gives us the following corollary.…”

“…The proof in [12] was based on an explicit combinatorial description of the roof function, and hands-on pressure estimates. That argument has the advantage of giving explicit and constructive examples for which uniqueness of the MME fails in the class of topological suspension flows.…”

The weak specification property for geodesic flows on $\mathrm{CAT}(–1)$ spaces

“…It is possible to construct a suspension flow over the full shift with more than one measure of maximal entropy, which rules out the possibility that this flow has weak specification. This construction is given in [32].…”

Measures of maximal entropy on subsystems of topological suspension semi-flows

“…In the current paper, we remove the restriction that the subset Y is a shift of finite type, and we remove the need for the ambient space to be symbolic, allowing any topological dynamical system with upper-semicontinuous entropy map. The proof in [12] was based on an explicit combinatorial description of the roof function, and hands-on pressure estimates. That argument has the advantage of giving explicit and constructive examples for which uniqueness of the MME fails in the class of topological suspension flows.…”