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.
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We consider two characteristic exponents of a rational function f:ℂ̂ → ℂ̂ of degree d ≥ 2. The exponent $χ_a(f)$ is the average of log∥f'∥ with respect to the measure of maximal entropy. The exponent $χ_m(f)$ can be defined as the maximal characteristic exponent over all periodic orbits of f. We prove that $χ_a(f) = χ_m(f)$ if and only if f(z) is conformally conjugate to $z ↦ z^{±d}$.
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Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space $ℙ^{k}$, 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 $κ_{f} > 0$ such that if ϕ: J → ℝ is a Hölder continuous function with $sup(ϕ) - inf(ϕ) < κ_{f}$, then ϕ admits a unique equilibrium state $μ_{ϕ}$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f,μ_{ϕ})$ is K-mixing, whence ergodic. Proving almost periodicity of the corresponding Perron-Frobenius operator is the main technical task of the paper. It requires producing sufficiently many "good" inverse branches and controling the distortion of the Birkhoff sums of the potential ϕ. In the case when the Julia set J does not intersect any periodic irreducible algebraic variety contained in the critical set of f, we have $κ_{f} = log d$, where d is the algebraic degree of f.
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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.
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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) $lim_{n→∞} n^{-1} log ∑_{j=0}^{τₙ(x)} μ(αⁿ(T^j(x)))$, where $αⁿ(T^j(x))$ is the element of the partition containing $T^j(x)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).
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