On the Regularity of the Multifractal Spectrum of Bernoulli Convolutions

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In previous work we developed a thermodynamic formalism for the Bernoulli convolution associated with the golden mean, and we obtained by perturbative analysis the existence, regularity, and strict convexity of the pressure F(β) in a neighborhood of β=0. This gives the existence of a multifractal spectrum ƒ(α) in a neighborhood of the almost sure value α=ƒ(α)=0, 9957…. In the present paper, by a direct study of the Ruelle–Perron–Frobenius operator associated with the random unbounded matrix product arising in our problem, we can prove the regularity of the pressure F(β) for (at least) β∈(−1/2,+∞). This yields the interval of the singularity spectrum between the minimal value of the dimension of v, αmin=0.94042…, and the almost sure value, αa.s.=0.9957….

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