The Pressure Function for Products of Non-negative Matrices

2002/05/02 by De-Jun Feng, De‐Jun Feng, Feng, De-Jun +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Chaos control and synchronization #Mathematical Dynamics and Fractals #math.CA #math.DS #msc:28A78 #msc:37D35

paper · pdf · doi:10.48550/arxiv.math/0205028

arxiv created 2002/05/02 · arxiv updated 2009/11/30

Abstract

Let (Σ_A, σ) be a subshift of finite type and let M(x) be a continuous function on Σ_A taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where M(x) takes finite values M₁, ..., M_m. The pressure function reduces to P(q):=lim_{n→ ∞}(1)/(n) log ∑_J ∈ ∑_{A, n} ‖M_J‖^q. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps.

Citations

• Products of Random Matrices

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