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Characteristic Exponents of Rational Functions

Anna Zdunik — 2014

Bulletin of the Polish Academy of Sciences. Mathematics

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 .

Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique

Feliks PrzytyckiAnna Zdunik — 1994

Fundamenta Mathematicae

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.

Equilibrium measures for holomorphic endomorphisms of complex projective spaces

Mariusz UrbańskiAnna Zdunik — 2013

Fundamenta Mathematicae

Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space 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 s u p ( ϕ ) - i n f ( 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...

Pressure and recurrence

Véronique Maume-DeschampsBernard SchmittMariusz UrbańskiAnna Zdunik — 2003

Fundamenta Mathematicae

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) l i m n n - 1 l o g j = 0 τ ( x ) μ ( α ( T j ( x ) ) ) , where α ( 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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