Ergodic properties of Erd\"os measure, the entropy of the goldenshift, and related problems

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 $\Bbb T^2$ 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.