Related papers: Hausdorff dimension OF dynamical Dophantine approx…
Let $(X,\mathscr{B}, \mu,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $\mu$ be an Ahlfors $s$-regular probability measure. The dynamical covering problem concerns the set $E(x)$ of points which…
In this article, we establish optimality results regarding the dynamical Borel-Cantelli lemma and the the Hausdorff dimension of certain dynamical diophantine sets.
We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of…
Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$, and let $U$ be a subset of $X$ whose complement is compact. We use the exponential mixing results for diagonalizable flows on $X$ to give upper estimates for the…
We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_n + \xi_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear…
We construct a self-affine sponge in $\mathbb R^3$ whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem…
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff…
We show some results about the Hausdorff dimension of particular minimal but not uniquely ergodic interval exchange transformations. There is an appendix which shows that typical points for two different ergodic measures of an interval…
We describe classes of ergodic dynamical systems for which some statistical properties are known exactly. These systems have integer dimension, are not globally dissipative, and are defined by a probability density and a two-form. This…
In this paper we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters…
We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is…
We study the size of \emph{dynamical covering sets} on a self-similar set. Dynamical covering sets are limsup sets generated by placing shrinking target sets around points along an orbit in a dynamical system. In the case when the target…
Using the variational principle in parametric geometry of numbers, we compute the Hausdorff and packing dimension of Diophantine sets related to exponents of Diophantine approximation, and their intersections. In particular, we extend a…
We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on…
Under the assumption of a natural subadditive potential, the so called cylinder function, working on the symbol space we prove the existence of the ergodic invariant probability measure satisfying the equilibrium state. As an application we…
We show the existence of a bounded Borel measurable saturated compensation function for a factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set…
In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including $A$-coupled expanding systems. We prove that Li-Yorke pairs of $A$- coupled-expanding system under some conditions have…
We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allows us to bound from above the…
LLet $f$ be a holomorphic endomorphism of $\mathbb P^ 2$ of degree $d \geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $\nu$. We infer several applications. The…
We fill a gap in the study of the Hausdorff dimension of the set of exact approximation order considered by Fregoli [Proc. Amer. Math. Soc. 152 (2024), no. 8, 3177--3182].