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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 are covered by the orbits of $xin X$ infinitely many times. We prove that the Hausdorff dimension of the intersection of $E(x)$ and any regular fractal $G$ equals $dim_{rm H}G+alpha-s$, where $alpha=dim_{rm H}E(x)$ $mu$--a.e. Moreover, we obtain the packing dimension of $E(x)cap G$ and an estimate for $dim_{rm H}(E(x)cap G)$ for any analytic set $G$.
We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without such a hypot
Nonstandard ergodic averages can be defined for a measure-preserving action of a group on a probability space, as a natural extension of classical (nonstandard) ergodic averages. We extend the one-dimensional theory, obtaining L^1 pointwise ergodic t
We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, ErdH
Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$breve{C}$ech compactification
We propose to compute approximations to general invariant sets in dynamical systems by minimizing the distance between an appropriately selected finite set of points and its image under the dynamics. We demonstrate, through computational experiments