Dimension Studies Of Liminf Sets In ?-expansions

The asymptotic properties of orbits are the core content of dynamical systems.Since the beginning of this century,people have done a lot of researches on the Limsup sets of ?-transformation dynamical systems,such as shrinking target problems and covering problems,but the results of the Liminf sets are still relatively few.In this paper,we study the dimension theory of the Liminf sets in ?-expansions,and focus on the distal property,we get the following two results:1)We completely describe the measure and dimension results of the sets of points which have different motion forms from the orbit of a given point.Let y ?[0,1),we consider the following decomposition(?),where(?) namely the set of distal pairs of y.AS?(y)is the set of asymptotic pairs of y,LY?(y)is the set of Li-Yorke pairs of y.We give the Lebesgue measures and Hausdorff dimensions of these three sets,for example,the Hausdorff dimension of set D?(y)is always full.We generalize the result along two different ways.On the one hand,we construct a appropriate subset of D?(y),and then prove that D?(y)is winning set by applying the approximation method in ?-expansions.On the other hand,we study a broader set(?) where {Yn}n?0(?)[0,1),that is to say,D?({yn})is the set of points which avoids a series of points.When yn=T?n(y),D?({yn})=D?(y).Moreover,we prove that the Hausdorff dimension of set D?({yn})is always full.2)We obtain the Lebesgue measure and Hausdorff dimension of the two-dimensional set of distal pairs.Spacing shifts systems are a subshift of symbolic space,we get the dimension results of the badly approximable set and the distal set of a given point.