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Parabolic Cantor sets

Mariusz Urbański — 1996

Fundamenta Mathematicae

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...

Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics

Mariusz Urbański — 2003

Fundamenta Mathematicae

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...

The box-counting dimension for geometrically finite Kleinian groups

B. StratmannMariusz Urbański — 1996

Fundamenta Mathematicae

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...

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...

Geometry of Markov systems and codimension one foliations

Andrzej BiśMariusz Urbański — 2008

Annales Polonici Mathematici

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.

Diophantine approximation in Banach spaces

Lior FishmanDavid SimmonsMariusz Urbański — 2014

Journal de Théorie des Nombres de Bordeaux

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.

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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