Let J be the Julia set of a conformal dynamics f . Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out...

Let β > 1 be a non-integer. We consider β-expansions of the form ∑∞ i=1 di βi , where the digits (di)i≥1 are generated by means of a random map Kβ defined on {0, 1}N× [0, bβc/(β − 1)]. We show that Kβ has a unique measure νβ of maximal entropy log(1 + bβc). Under this measure, the digits (di)i≥1 form a uniform Bernoulli process, and the projection of this measure on the second coordinate is an ...

In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, prove: m.m.e. is Bernoulli, and periodic points with Lyapunov exponents bounded away from zero equidistribute respect to m.m.e.We some estimates regarding Hausdorff dimension about density support on manifold. For generic parameter, 2. We also obtain $C^2$-r...