Abstract
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 f(α) in a neighborhood of the almost sure value α=f(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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Porzio, A. On the Regularity of the Multifractal Spectrum of Bernoulli Convolutions. Journal of Statistical Physics 91, 17–29 (1998). https://doi.org/10.1023/A:1023027718308
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DOI: https://doi.org/10.1023/A:1023027718308