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Ergodic properties of the Erdős measure, the entropy of the goldenshift, and related problems. (English) Zbl 0916.28012

The Erdős measure on the interval [P. Erdős, Am. J. Math. 61, 974-976 (1939; Zbl 0022.35402)] is the singular measure corresponding to the distribution of the random variable \(\sum_{n=1}^{\infty}\varepsilon_n\rho^{-n}\) where \(\rho\) is the golden mean and \(\varepsilon_n\) is a sequence of i.i.d. random variables taking the values 0 and 1 with equal probability. In this extended study, the authors introduce a two-sided generalization of the Erdős measure and a map that preserves this measure (the ‘goldenshift’). The entropy is computed and related to the pointwise dimension of the Erdős measure. The work applies ideas from the work of A. M. Vershik on arithmetical expansions and dynamics [Funkts. Anal. Prilozh. 26, No. 3, 22-27 (1992; Zbl 0810.58031)].

MSC:

28D05 Measure-preserving transformations
28D20 Entropy and other invariants
37E99 Low-dimensional dynamical systems
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References:

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