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Let f be a polynomial of one complex variable so that its Julia set is connected. We show that the harmonic (Brolin) measure of the set of biaccessible points in J is zero except for the case when J is an interval.

We consider two characteristic exponents of a rational function f:ℂ̂ → ℂ̂ of degree d ≥ 2. The exponent ${\chi }_a\left(f\right)$ is the average of log∥f’∥ with respect to the measure of maximal entropy. The exponent ${\chi }_m\left(f\right)$ can be defined as the maximal characteristic exponent over all periodic orbits of f. We prove that ${\chi }_a\left(f\right)={\chi }_m\left(f\right)$ if and only if f(z) is conformally conjugate to $z\mapsto z^{\pm d}$.

We prove that if A is a basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then the periodic points in the boundary of A are dense in this boundary. To prove this in the non-simply connected or parabolic situations we prove a more abstract, geometric coding trees version.

Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space ${\mathbb{P}}^k$, k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number ${\kappa }_f>0$ such that if ϕ: J → ℝ is a Hölder continuous function with $sup\left(\varphi \right)-inf\left(\varphi \right)<{\kappa }_f$, then ϕ admits a unique equilibrium state ${\mu }_{\varphi }$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f,{\mu }_{\varphi })$ is K-mixing, whence ergodic. Proving...

We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) $li{m}_{n\to \infty }{n}^{-1}log{\sum }_{j=0}^{\tau ₙ\left(x\right)}\mu \left(\alpha ⁿ\left(T^j\left(x\right)\right)\right)$, where $\alpha ⁿ\left(T^j\left(x\right)\right)$ is the element of the partition containing $T^j\left(x\right)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).