Related papers: Bowen Parameter and Hausdorff Dimension for Expand…

Suppose $X$ is a compact connected metric space and $f: X \to X$ is a metric coarse expanding conformal map in the sense of Ha\"issinsky-Pilgrim. We show that if $X$ contains a homeomorphic copy of the letter "Y", then the Hausdorff…

This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular…

Let $\mathbb{H}$ be the sub-Riemannian Heisenberg group. That $\mathbb{H}$ supports a rich family of quasiconformal mappings was demonstrated by Kor\'{a}nyi and Reimann using the so-called flow method. Here we supply further evidence of the…

One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups…

The first result of the paper (Theorem 1.1) is an explicit construction of unimodal maps that are semiconjugate, on the post-critical set, to the circle rotation by an arbitrary irrational angle $\theta\in(3/5,2/3)$. Our construction is a…

We study families of analytic semigroups, acting in a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective…

We present a dynamical and spectral study of Ostrowski's map based on the use of transfer operators. The Ostroswki dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and interval fibers) and…

From a geometric perspective, we employ metric mean dimension to investigate the set of generic points of invariant measures and saturated sets in infinite entropy systems. For systems with the specification property, we establish certain…

We consider one-dimensional difference Schroedinger equations on the discrete line with a potential generated by evaluating a real-analytic potential function V(x) on the one-dimensional torus along an orbit of the shift x-->x+nw. If the…

Let c be a real parameter in the Mandelbrot set, and f_c(z):= z^2 + c. We prove a formula relating the topological entropy of f_c to the Hausdorff dimension of the set of rays landing on the real Julia set, and to the Hausdorff dimension of…

A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the…

In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner…

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their…

Let f be a dominant rational map of P^k such that there exists s <k, with lambda_s(f)>lambda_l(f) for all l. Under mild hypotheses, we show that, for A outside a pluripolar set of the group of automorphisms of P^k, the map f o A admits a…

For every finite generating set on the integer Heisenberg group H(Z), Pansu showed that the word metric has the large-scale structure of a Carnot-Caratheodory Finsler metric on the real Heisenberg group H(R). We study the properties of…

We prove that the boundary of a component $U$ of the basin of an attracting periodic cycle (of period greater than 1) for an exponential map on the complex plane has Hausdorff dimension greater than 1 and less than 2. Moreover, the set of…

We consider an infinite extension $K$ of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. $K$ is equipped with an inductive limit topology; its conjugate $\bar{K}$ is a completion of $K$…

We consider the set of points with high pointwise emergence for $C^{1+\alpha}$ diffeomorphisms preserving a hyperbolic measure. We find a lower bound on the Hausdorff dimension of this set in terms of unstable Hausdorff dimension of the…

Belinschi and Nica introduced a composition semigroup on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know whether a probability measure is freely infinitely…

We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random $3$-regular graphs and making using of the Ollivier curvature, a coarse analogue of the Ricci curvature. Numerical analysis shows that…