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

Marc Kesseböhmer; Sabrina Kombrink

doi:10.3934/dcdss.2017016

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2017, Volume 10, Issue 2: 335-352. Doi: 10.3934/dcdss.2017016

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

Marc Kesseböhmer^1, and
Sabrina Kombrink^2,

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: November 30, 2015

Revised: October 31, 2016

Published: February 2018

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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.

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Mathematics Subject Classification: 58F11 (28D99, 58F03).

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References

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