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Parabolic Cantor sets

100%
Urbański M.
Fundamenta Mathematicae
|
1996
|
tom 151
|
nr 3
241-277
EN
The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional packing measure is positive and finite. This latter measure can also be dynamically characterized as the only h-conformal measure. It is relatively easy to see that any two parabolic Cantor sets formed with the help of the same alphabet are canonically topologically conjugate and we then discuss the rigidity problem of what are the possibly weakest sufficient conditions for this topological conjugacy to be "smoother". It turns out that if the conjugating homeomorphism preserves the moduli of the derivatives at periodic points, then the dimensions of both sets are equal and the homeomorphism is shown to be absolutely continuous with respect to the corresponding h-dimensional packing measures. This property in turn implies the conjugating homeomorphism to be Lipschitz continuous. Additionally the existence of the scaling function is shown and a version of the rigidity theorem, expressed in terms of scaling functions, is proven. We also study the real-analytic Cantor sets for which the stronger rigidity can be shown, namely that the absolute continuity of the conjugating homeomorphism alone implies its real analyticity.
2
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Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics

100%
Urbański M.
Fundamenta Mathematicae
|
2003
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tom 176
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nr 2
97-125
EN
We show that for critically non-recurrent rational functions all the definitions of topological pressure proposed in [12] coincide for all t ≥ 0. Then we study in detail the Gibbs states corresponding to the potentials -tlog|f'| and their σ-finite invariant versions. In particular we provide a sufficient condition for their finiteness. We determine the escape rates of critically non-recurrent rational functions. In the presence of parabolic points we also establish a polynomial rate of appropriately modified escape. This extends the corresponding result from [6] proven in the context of parabolic rational functions. In the last part of the paper we introduce the class of critically tame generalized polynomial-like mappings. We show that if f is a critically tame and critically non-recurrent generalized polynomial-like mapping and g is a Hölder continuous potential (with sufficiently large exponent if f has parabolic points) and the topological pressure satisfies P(g) > sup(g), then for sufficiently small δ >0, the function t↦ P(tg), t ∈ (1-δ,1+δ), is real-analytic.
3
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The box-counting dimension for geometrically finite Kleinian groups

64%
Stratmann B., Urbański M.
Fundamenta Mathematicae
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1996
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tom 149
|
nr 1
83-93
EN
We calculate the box-counting dimension of the limit set of a general geometrically finite Kleinian group. Using the 'global measure formula' for the Patterson measure and using an estimate on the horoball counting function we show that the Hausdorff dimension of the limit set is equal to both: the box-counting dimension and packing dimension of the limit set. Thus, by a result of Sullivan, we conclude that for a geometrically finite group these three different types of dimension coincide with the exponent of convergence of the group.
4
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Equilibrium measures for holomorphic endomorphisms of complex projective spaces

64%
Urbański M., Zdunik A.
Fundamenta Mathematicae
|
2013
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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.
5
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Geometry of Markov systems and codimension one foliations

64%
Biś A., Urbański M.
Annales Polonici Mathematici
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2008
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tom 94
|
nr 2
187-196
EN
We show that the theory of graph directed Markov systems can be used to study exceptional minimal sets of some foliated manifolds. A C¹ smooth embedding of a contracting or parabolic Markov system into the holonomy pseudogroup of a codimension one foliation allows us to describe in detail the h-dimensional Hausdorff and packing measures of the intersection of a complete transversal with exceptional minimal sets.
6
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Conformal measures for rational functions revisited

51%
Denker M., Daniel Mauldin R., Nitecki Z., Urbański M.
Fundamenta Mathematicae
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1998
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tom 157
|
nr 2-3
161-173
EN
We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.
7
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Pressure and recurrence

51%
Maume-Deschamps V., Schmitt B., Urbański M., Zdunik A.
Fundamenta Mathematicae
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2003
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tom 178
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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).
8
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On the uniqueness of equilibrium states for piecewise monotone mappings

51%
Denker M., Keller G., Urbański M.
Studia Mathematica
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1990
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tom 97
|
nr 1
27-36
9
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Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II

51%
Przytycki F., Urbański M., Zdunik A.
Studia Mathematica
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1990
|
tom 97
|
nr 3
189-225
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