Abstract
We deal with countable alphabet locally compact random subshifts of finite type (the latter merely meaning that the symbol space is generated by an incidence matrix) under the absence of big images property and under the absence of uniform positivity of the transfer operator. We first establish the existence of random conformal measures along with good bounds for the iterates of the Perron–Frobenius operator. Then, using the technique of positive cones and proving a version of Bowen’s type contraction (see Equilibrium states and the ergodic theory of Anosov diffeomorphisms, 1975), we also establish a fairly complete thermodynamical formalism. This means that we prove the existence and uniqueness of fiberwise invariant measures (giving rise to a global invariant measure) equivalent to the fiberwise conformal measures. Furthermore, we establish the existence of a spectral gap for the transfer operators, which in the random context precisely means the exponential rate of convergence of the normalized iterated transfer operator. This latter property in a relatively straightforward way entails the exponential decay of correlations and the Central Limit Theorem.
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Acknowledgments
We wish to thank the anonymous referee for valuable remarks and suggestions which improved the final exposition of our paper. We particularly thank him or her for bringing up to our attention the distinction between annealed and quenched results. Mariusz Urbański was supported in part by the NSF Grant DMS 1361677.
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Mayer, V., Urbański, M. Countable Alphabet Random Subhifts of Finite Type with Weakly Positive Transfer Operator. J Stat Phys 160, 1405–1431 (2015). https://doi.org/10.1007/s10955-015-1287-9
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DOI: https://doi.org/10.1007/s10955-015-1287-9
Keywords
- Random dsynamical systems
- Thermodynamical formalism
- Exponential decay of correlations
- Central limit theorem
- Markov shifts