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Register a new userMeasures of maximal entropy for suspension flows over the full shift
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We consider suspension flows with continuous roof function over the full shift $Sigma$ on a finite alphabet. For any positive entropy subshift of finite type $Y subset Sigma$, we explictly construct a roof function such that the measure(s) of maximal entropy for the suspension flow over $Sigma$ 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 Holder continuous roof function where it is well known the measure of maximal entropy is unique and fully supported.
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Given a compact topological dynamical system (X, f) with positive entropy and upper semi-continuous entropy map, and any closed invariant subset $Y subset X$ with positive entropy, we show that there exists a continuous roof function such that the set of measures of maximal entropy for the suspension semi-flow over (X,f) consists precisely of the lifts of measures which maximize entropy on Y. This result has a number of implications for the possible size of the set of measures of maximal entropy for topological suspension flows. In particular, for a suspension flow on the full shift on a finite alphabet, the set of ergodic measures of maximal entropy may be countable, uncountable, or have any finite cardinality.
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