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...

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...

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...

Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space ${\mathbb{P}}^{k}$, 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 ${\kappa}_{f}>0$ such that if ϕ: J → ℝ is a Hölder continuous function with $sup\left(\varphi \right)-inf\left(\varphi \right)<{\kappa}_{f}$, then ϕ admits a unique equilibrium state ${\mu}_{\varphi}$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f,{\mu}_{\varphi})$ is K-mixing, whence ergodic. Proving...

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.

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.

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.

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) $li{m}_{n\to \infty}{n}^{-1}log{\sum}_{j=0}^{\tau \u2099\left(x\right)}\mu \left(\alpha \u207f\left({T}^{j}\left(x\right)\right)\right)$, where $\alpha \u207f\left({T}^{j}\left(x\right)\right)$ is the element of the partition containing ${T}^{j}\left(x\right)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

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