Normalized image of a vector by an infinite product of nonnegative\n matrices

2022/05/17 by Alain Thomas, Thomas, Alain
Mathematics · Physics and Astronomy · #15B48 #28A12 #FOS: Mathematics #Functional Analysis (math.FA) #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2205.09044

openalex publication_date 2022/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To prove that a measure, linearly representable by means of a finite set of\nnonnegative matrices mathcal M, has the weak-Gibbs property, one check the\nuniform convergence (on mathcal M^ mathbb N) of the sequence of vectors\n\(A₁\⋯ A_nc)/( Vert A₁\⋯ A_nc Vert) (c positive\ncolumn-vector). The main theorem gives a sufficient condition for this sequence\nto converge pointwise. This theorem generalizes the Birkhoff contraction method\nbecause it can be used even if the matrices have many zero entries. We also\nlook at the convergence of the sequence of matrices \(A₁\⋯ A_n)/( Vert\nA₁\⋯ A_n Vert). The measures defined by Bernoulli convolution are in\ncertain cases linearly representable; we give two example of weak-Gibbs\nBernoullt convolutions, by using the Birkhoff contraction coefficient for the\nfirst and the theorem for the second. Furthermore we explicit the relationship\nbetween the notions of Bernoulli convolution, fundamental curves and lattice\ntwo-scale difference equations.\n

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