A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory
Mark Kesseb\"ohmer, Sabrina Kombrink
TL;DR
This paper extends the Ruelle-Perron-Frobenius theorem to infinite alphabet Markov shifts and applies it to broaden renewal theory in symbolic dynamics, enabling analysis in more complex systems.
Contribution
It introduces a complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts, generalizing prior finite alphabet results and extending renewal theory to infinite symbolic systems.
Findings
Extended Ruelle-Perron-Frobenius theorem to infinite alphabets
Generalized renewal theory in symbolic dynamics
Applicable to complex systems with infinite states
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.
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