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Register a new userWeak Gibbs and Equilibrium Measures for Shift Spaces
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Added by
Charles-Edouard Pfister
Publication date
2019
fields
Physics
and research's language is
English
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For a large class of irreducible shift spaces $XsubsettA^{Z^d}$, with $tA$ a finite alphabet, and for absolutely summable potentials $Phi$, we prove that equilibrium measures for $Phi$ are weak Gibbs measures. In particular, for $d=1$, the result holds for irreducible sofic shifts.
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Let (X,T) be a dynamical system, where X is a compact metric space and T a continuous onto map. For weak Gibbs measures we prove large deviations estimates.
We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Holder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates.
This paper has been withdrawn by the authors due to an error in the main theorem.
Let $G=leftlangle S|R_{A}rightrangle $ be a semigroup with generating set $ S$ and equivalences $R_{A}$ among $S$ determined by a matrix $A$. This paper investigates the complexity of $G$-shift spaces by yielding the topological entropies. After revealing the existence of topological entropy of $G$-shift of finite type ($G$-SFT), the calculation of topological entropy of $G$-SFT is equivalent to solving a system of nonlinear recurrence equations. The complete characterization of topological entropies of $G$-SFTs on two symbols is addressed, which extends [Ban and Chang, arXiv:1803.03082] in which $G$ is a free semigroup.
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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