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On biaccessible points in Julia sets of polynomials

100%
Zdunik A.
Fundamenta Mathematicae
|
2000
|
tom 163
|
nr 3
277-286
EN
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.
2
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Characteristic Exponents of Rational Functions

100%
Zdunik A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2014
|
tom 62
|
nr 3
257-263
EN
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 $z ↦ z^{±d}$.
3
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Equilibrium measures for holomorphic endomorphisms of complex projective spaces

64%
Urbański M., Zdunik A.
Fundamenta Mathematicae
|
2013
|
tom 220
|
nr 1
23-69
EN
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.
4
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Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique

64%
Przytycki F., Zdunik A.
Fundamenta Mathematicae
|
1994
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tom 145
|
nr 1
65-77
EN
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.
5
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Pressure and recurrence

51%
Maume-Deschamps V., Schmitt B., Urbański M., Zdunik A.
Fundamenta Mathematicae
|
2003
|
tom 178
|
nr 2
129-141
EN
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).
6
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Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II

51%
Przytycki F., Urbański M., Zdunik A.
Studia Mathematica
|
1990
|
tom 97
|
nr 3
189-225
7
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Entropy of transformations of the unit interval

32%
Zdunik A.
Fundamenta Mathematicae
|
1984
|
tom 124
|
nr 3
235-241
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