Fibonacci Adding Machine and Filled Julia Set

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Date(s) - 09/09/2014
11 h 00 min - 12 h 00 min

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In this work we consider a stochastic adding machine associated to the Fibonacci sequence. We obtain a Markov chain with states in N and we study some properties of the spectrum of the transition operator (defined in l∞) associated to this Markov chain. In particular, we prove that the spectrum of the transition operator associated to this Markov chain is related with the filled Julia set K = {(x, y) ∈ ℂ² : fn◦fn₋₁◦…◦f₁(x, y) is bounded}, where fᵢ are maps defined in ℂ² by fᵢ(x, y) = (xy + cᵢ, x), and cᵢ ∈ ℝ, for all i ∈ N. More precisely, we prove that the spectrum is equal to the set S = {z ∈ ℂ : (z, z) ∈ K}. We also show some topological properties of S.

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Olivier CHABROL
Posts created 14

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