Dimension Theory Of Three Kinds Of Limsup Sets And Their Related Problems

Limsup sets are defined as the upper limit of a sequence of sets,which plays an important role in different mathematical fields,such as well approximated sets in Diophantine approximation,shrinking target sets in dynamic systems,and thick points of Brownian motion in stochastic processes.We considered Hausdorff dimensions of recurrence sets,limsup random fractals and random covering sets,and related problems.The study of recurrence sets originates from the Poincare recurrence theorem.Currently,Lots of related studies focus on expanding dynamical systems.We considered linear systems(T2,T)on the 2-dimensional torus T2 where expansion and compression coexists,and T is an integer matrix transformation on T2.We gave a formula for the Hausdorff dimension of the recurrence sets R={x∈Td:x∈B(Tn(x).rn)for infinitely many n},where {rn}n≥1 is a sequence of positive real numbers decreasing to 0,and B(Tn(x),rn)denotes the open ball with centre Tnx,and radius rn.At the same time,we also proved that R has large intersection property.Limsup random fractals are closely related to fractal sets in many stochastic processes.Let A be a limsup random fractal in a compact metric space(X,d).We studied the intersection problem of limsup random fractals with fractals.For any analytic set G?X,we provided criteria for determining whether A∩G is empty or not,and calculated the Hausdorff dimension of A∩G when G is regular.The conclusions above cover the results in Euclidean space,and we also used the hitting probabilities of limsup random fractals to investigate the intersection problem of random covering sets with fractals.There are lots of outstanding authors who pay much attention on the classic Dovertzky covering problem,such as Billard(1965)、Erd?s(1961)、Kahane(1985)、Mandelbrot(1972),Shepp(1972)and so on.The problem concerns independent and uniformly distributed random variables on the torus,but we considered the random covering sets without independence.Let {ξn}n be an exponentially mixing stationary process.We studied the set from the viewpoint of measure,dimension and topology.We also considered the hitting probability of random covering sets,and we obtained the Hausdorff dimension oi E∩G when G is regular.Applying these results to dynamical systems,we used methods in probability theory to deal with the intersection problem in dynamical systems,and obtained the Hausdorff dimension of the intersection of dynamical covering sets and regular fractals in the exponentially mixing measure preserving systems.