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Gibbs states on the symbolic space over an infinite alphabet

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Abstract

We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Hölder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

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Authors and Affiliations

  1. Department of Mathematics, University of North Texas, 76203-1430, Denton, TX, USA

    R. Daniel Mauldin & Mariusz Urbański

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  1. R. Daniel Mauldin
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  2. Mariusz Urbański
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Correspondence to R. Daniel Mauldin.

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Partially supported by NSF Grant DMS 9801583.

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Mauldin, R.D., Urbański, M. Gibbs states on the symbolic space over an infinite alphabet. Isr. J. Math. 125, 93–130 (2001). https://doi.org/10.1007/BF02773377

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  • DOI: https://doi.org/10.1007/BF02773377

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