Related papers: Continuity of Hausdorff dimension across generic d…

Let $\varphi_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $\Lambda_0$ be a transitive horseshoe of $\varphi_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation…

Let $\varphi_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let $\Lambda_0$ be a mixing horseshoe of $\varphi_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $\varphi$, close to…

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface $N$, and let $\Lambda_0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small…

Let $\varphi$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $\Lambda$ be a mixing horseshoe of $\varphi$. Given a smooth real function $f$ defined on $S$, we define for points $\eta$ in the unstable Cantor set of…

We consider the Lagrange and the Markov dynamical spectra associated to horseshoes on surface with Hausdorff dimensions greater than 1. We show that for a "large" set of real functions on the surface and for "typical" horseshoes with…

We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence…

This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem 2.1, provides a rigorous estimate on the smallest value $t_1$ such that the portion…

For a C^{1+\alpha} diffeomorphism f preserving a hyperbolic ergodic SRB measure \mu, Katok's remarkable results assert that \mu can be approximated by a sequence of hyperbolic sets \{\Lambda_n\}_{n\geq1}. In this paper, we prove the…

In this note we derive an upper bound for the Hausdorff dimension of the stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that $\dim_H W^s(\Lambda)=n$ is…

Let $f: M\rightarrow M$ be a continuous map on a compact metric space $M$ equipped with a fixed metric $d$, and let $\tau$ be the topology on $M$ induced by $d$. First, we will establish some fundamental properties of the mean Hausdorff…

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $\Lambda(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in \Lambda(\mathbf{f}):…

For a hyperbolic map f on a saddle type fractal Lambda with self-intersections, the number of f- preimages of a point x in Lambda may depend on x. This makes estimates of the stable dimensions more difficult than for diffeomorphisms or for…

For a non-conformal repeller $\Lambda$ of a $C^{1+\alpha}$ map $f$ preserving an ergodic measure $\mu$ of positive entropy, this paper shows that the Lyapunov dimension of $\mu$ can be approximated gradually by the Carath\'{e}odory singular…

The Markov and Lagrange dynamical spectra, was introduced by Moreira and share several geometric and topological aspects with the classical ones. However, some features of generic dynamical spectra associated to hyperbolic sets can be…

It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs.…

In this paper, we show that geometric Lorenz attractors have Hausdorff dimension strictly greater than $2$. We use this result to show that for a "large" set of real functions the Lagrange and Markov Dynamical spectrum associated to these…

Suppose that $\Omega = \{0, 1\}^ {\mathbb {N}}$ and $ {\sigma}$ is the one-sided shift. The Birkhoff spectrum $ \displaystyle S_{f}( {\alpha})=\dim_{H}\Big \{ {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(\sigma^n…

Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian…

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is…

We explicitly calculate the Hausdorff dimension of the graph and range of an isotropic stable L\'{e}vy process $X$ plus deterministic drift function $f$. For that purpose we use a restricted version of the genuine Hausdorff dimension which…