QUASI-COMPACTNESS OF TRANSFER OPERATORS FOR TOPOLOGICAL MARKOV SHIFTS WITH HOLES

. We consider transfer operators for topological Markov shift (TMS) with countable states and with holes which are 2-cylinders. We found that, if the closed system of the shift had an irreducible transition matrix and the potential was a weaker Lipschitz continuous and summable, then we obtained a version of Ruelle-Perron-Frobenius theorem and quasi-compactness of the associated Ruelle transfer operator. The escape rate of the open system was also calculated. We also found that the Ruelle operator of summable potential on topologically transitive TMS has a spectral gap property. As another example, we applied the main results to the transfer operators associated to graph iterated function systems.