  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.
