Abstract
We consider a perturbed system (\(\Sigma_A^+\),ϕ(ϵ, ·)) with topologically transitive subshift of finite type \(\Sigma_A^+\) and Hölder continuous functions ϕ(ϵ, ·) defined on \(\Sigma_A^+\) endowed with small parameter ϵ > 0. Through our choice of ϕ(ϵ, ·), we realize the situation that the perturbed system has a unique Gibbs measure μϵ of the potential ϕ(ϵ, ·) for each ϵ > 0 and on the other hand the unperturbed system possesses several Gibbs measures μ1, μ2,…, μm of the limit potential at ϵ = 0. In this paper, we give a necessary and sufficient condition for convergence of the measure μϵ using the notion of Perron complements of Ruelle operators. Our results can be applied also to the problems of convergence of stationary distributions of perturbed Markov chains with holes.
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Tanaka, H. Perturbed finite-state Markov systems with holes and Perron complements of Ruelle operators. Isr. J. Math. 236, 91–131 (2020). https://doi.org/10.1007/s11856-020-1968-1
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DOI: https://doi.org/10.1007/s11856-020-1968-1