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Register a new userOn the Intersection of Dynamical Covering Sets with Fractals
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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 $G$.
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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 hypothesis). Next we prove that, in general, such an intersection consists of a nonsingular hyperbolic set, finitely many singularities and regular orbits joining them. Afterward we exhibit a three-dimensional star flow with two homoclinic classes, one being positively (but not negatively) sectional-hyperbolic and the other negatively (but not positively) sectional-hyperbolic, whose intersection reduces to a single periodic orbit. This provides a counterexample to a conjecture by Shy, Zhu, Gan and Wen (cite{sgw}, cite{zgw}).
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 theorems for several kinds of nonstandard sparse group averages, with a special focus on the group Z^d. Namely, we extend results for sparse block averages and sparse random averages to their analogues on virtually nilpotent groups, and extend Christs result for sparse deterministic sequences to its analogue on Z^d. The second and third results have two nontrivial variants on Z^d: a native d-dimensional average and a product average from the 1-dimensional averages.
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{o}s, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Levy on the regular continued fraction expansion of irrational numbers in $(0,1)$.
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, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central theorem near zero. Algebraically one can define quasi-central set near zero for dense subsemigroup of $((0,infty),+)$, and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of $((0,infty),+)$, C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. Like discrete case, we shall produce dynamical characterizations of these combinatorically rich sets near zero.
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 that this approach can successfully converge to approximations of (maximal) invariant sets of arbitrary topology, dimension and stability as, e.g., saddle type invariant sets with complicated dynamics. We further propose to extend this approach by adding a Lennard-Jones type potential term to the objective function which yields more evenly distributed approximating finite point sets and perform corresponding numerical experiments.
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