# American Institute of Mathematical Sciences

April  2017, 10(2): 335-352. doi: 10.3934/dcdss.2017016

## A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory

 1 FB03-Mathematik und Informatik, Universität Bremen, Bibliothekstr. 1, 28359 Bremen, Germany 2 Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck, Germany

Received  December 2015 Revised  November 2016 Published  January 2017

We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.

Citation: Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016
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