Large Intersection Properties Research Articles

Path-dependent shrinking target problems in beta-dynamical systems

This paper is concerned with the path-dependent shrinking target problems in beta-dynamical systems. For β > 1 let T β be the β-transformation on . Let , and be two positive continuous functions. For any , we show that the set has large intersection property under the condition , where denotes the ergodic sum . Consequently, we can determine the Hausdorff dimension of , which is the solution to some modified pressure function defined through singular value function.

Large Intersection Property for Limsup Sets in Metric Space

We show that limsup sets generated by a sequence of open sets in compact Ahlfors s-regular $(0<s<\infty )$ space $(X,{\mathscr{B}},\mu ,\rho )$ belong to the classes of sets with large intersections with index λ, denoted by $\mathcal {G}^{\lambda }(X)$ , under some conditions. In particular, this provides a lower bound on Hausdorff dimension of such sets. These results are applied to obtain that limsup random fractals with indices γ2 and δ belong to $\mathcal {G}^{s-\delta -\gamma _{2}}(X)$ almost surely, and random covering sets with exponentially mixing property belong to $\mathcal {G}^{s_{0}}(X)$ almost surely, where s0 equals to the corresponding Hausdorff dimension of covering sets almost surely. We also investigate the large intersection property of limsup sets generated by rectangles in metric space.

Sets with large intersection properties in metric spaces

In this work we reproduce the characterization of Gs-sets from the euclidean setting [11] to more general metric spaces. These sets have Hausdorff dimension at least s and are closed by countable intersections, which is particularly useful to estimate the dimension of the so called sets of α-approximable points.

Highly irregular orbits for subshifts of finite type: large intersections and emergence

In their recent paper (2019 arXiv:1904.03424), the first author, Kiriki and Soma introduced a concept of pointwise emergence to measure the complexity of irregular orbits. They constructed a residual subset of the full shift with high pointwise emergence. In this paper we consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. We show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any Hölder continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property. This is a natural generalization of Färm and Persson (2011 Nonlinearity 24 1291–1309) to pointwise emergence and Carathéodory dimension.

On shrinking targets for piecewise expanding interval maps

For a map $T:[0,1]\rightarrow [0,1]$ with an invariant measure $\unicode[STIX]{x1D707}$, we study, for a $\unicode[STIX]{x1D707}$-typical $x$, the set of points $y$ such that the inequality $|T^{n}x-y|&lt;r_{n}$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D719}}$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

Describability via ubiquity and eutaxy in Diophantine approximation

We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of Levy processes.

A note on random coverings of tori

This note provides a generalisation of a recent result by J\"arvenp\"a\"a, J\"arvenp\"a\"a, Koivusalo, Li, and Suomala, (to appear), on the dimension of limsup-sets of random coverings of tori. The result in this note is stronger in the sense that it provides also a large intersection property of the limsup-sets, the assumptions are weaker, and it implies the result of J\"arvenp\"a\"a, J\"arvenp\"a\"a, Koivusalo, Li, and Suomala as a special case. The proof is based on a recent result by Persson and Reeve from 2013.

Non-Typical Points for β-Shifts

We study sets of non-typical points under the map fβ↦βx mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β>1.541 found in the above paper can be removed.

Large intersection classes on fractals

We consider limit sets of conformal iterated function systems, and introduce classes of subsets of these limit sets, with the property that the classes are closed under countable intersections and that all sets in the classes have a large Hausdorff dimension. Using these classes we determine the Hausdorff dimension and large intersection properties of some sets occurring in ergodic theory, Diophantine approximation and complex dynamics.

Multifractal analysis of Lévy fields

We study the pointwise regularity properties of the Lévy fields introduced by T. Mori; these fields are the most natural generalization of Lévy processes to the multivariate setting. We determine their spectrum of singularities, and we show that their Hölder singularity sets satisfy a large intersection property in the sense of K. Falconer.

Simultaneously non-dense orbits under different expanding maps

Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. It is well-known that in many cases such sets have full Hausdorff dimension. We prove that such sets have a large intersection property, i.e. countable intersections of such sets also have full Hausdorff dimension. This result applies to a class of maps including multiplication by integers modulo 1 and x ↦ 1/x modulo 1. We prove that the same properties hold for multiplication modulo 1 by a dense set of non-integer numbers between 1 and 2.

Large intersection properties in Diophantine approximation and dynamical systems

We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine approximation, in the study of the homeomorphisms of the circle and in the perturbation theory for Hamiltonian systems.

Ubiquity and large intersections properties under digit frequencies constraints

AbstractWe are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

Singularity sets of Lévy processes

We completely describe the size and large intersection properties of the Holder singularity sets of Levy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Levy process. The Holder singularity sets of the sample paths of certain random wavelet series are investigated as well.

Sets with large intersection and ubiquity

AbstractA central problem motivated by Diophantine approximation is to determine the size properties of subsets of$\R^d$ ($d\in\N$)of the formwhere ‖⋅‖ denotes an arbitrary norm,Ia denumerable set, (xi,ri)i∈ Ia family of elements of$\R^d\$× (0, ∞) and ϕ a nonnegative nondecreasing function defined on [0, ∞). We show that ifFId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subsetVof$\R^d\$, the setFϕbelongs to a class Gh(V) of sets with large intersection inVwith respect to a given gauge functionh. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorffg-measure for every gauge functiongwhich increases faster thanhnear zero. In particular, this yields a sufficient condition on a gauge functiongsuch that a given countable intersection of sets of the formFϕhas infinite Hausdorffg-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequenceψof positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that areψ-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarník's theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahler's and Koksma's classifications of real transcendental numbers.

Sets with Large Intersection Properties

Sets with Large Intersection Properties