Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

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

Pages: 335 - 352, Volume 10, Issue 2, April 2017      doi:10.3934/dcdss.2017016

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Marc Kesseböhmer - FB03 - Mathematik und Informatik, Universität Bremen, Bibliothekstr. 1, 28359 Bremen, Germany (email)
Sabrina Kombrink - Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck, Germany (email)

Abstract: 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.

Keywords:  Ruelle-Perron-Frobenius operator, renewal theory, infinite alphabet subshift.
Mathematics Subject Classification:  58F11 (28D99, 58F03).

Received: December 2015;      Revised: November 2016;      Available Online: January 2017.