Stochastic adding machine and $2$-dimensional Julia sets

Ali Messaoudi; Rafael Asmat Uceda

doi:10.3934/dcds.2014.34.5247

Article Contents

2014, Volume 34, Issue 12: 5247-5269. Doi: 10.3934/dcds.2014.34.5247

This issue Previous Article Some remarks on symmetric periodic orbits in the restricted three-body problem Next Article The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion

Stochastic adding machine and $2$-dimensional Julia sets

Ali Messaoudi^1, and
Rafael Asmat Uceda^2,

1.
UNESP - Departamento de Matemática do, Instituto de Biociências, Letras e Ciências, Exatas de São José do Rio Preto
2.
Université de Picardie Jules Verne, Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS - UMR 7352

Received: May 31, 2013

Revised: February 28, 2014

Published: November 2014

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Abstract

In this work we define a stochastic adding machine associated to a quadratic base $(F_n)_{n \geq 0}$ formed by recurrent sequences of order 2. We obtain a Markov chain with states in $\mathbb{Z}^+$ and we prove that the spectrum of the transition operator associated to this Markov chain is connected to the filled Julia sets for a class of endomorphisms in $\mathbb{C}^2$ of which we study topological properties.

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Mathematics Subject Classification: Primary: 37A30, 37F50; Secondary: 47A10.

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References

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