## Institute for Mathematical Sciences

## Preprint ims96-14

** N. Sidorov and A. Vershik**
* Egrodic Properties of Erd\"os Measure, the Entropy of the Goldenshift, and Related Problems.*

Abstract: We define a two-sided analog of Erd\"os measure on the space of two-sided expansions with respect to the powers of the golden
ratio, or, equivalently, the Erd\"os measure on the 2-torus. We
construct the transformation (goldenshift) preserving both
Erd\"os and Lebesgue measures on $T^2$ which is the
induced automorphism with respect to the ordinary shift (or the
corresponding Fibonacci toral automorphism) and proves to be
Bernoulli with respect to both measures in question. This
provides a direct way to obtain formulas for the entropy
dimension of the Erd\"os measure on the interval, its entropy
in the sense of Garsia-Alexander-Zagier and some other results.
Besides, we study central measures on the Fibonacci graph, the
dynamics of expansions and related questions.

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