Metric Diophantine Approximation In Dynamical Systems

(X,T)denotes a continuous dynamical system in the sense that T:X 4 X is a continuous transformation on the compact metric space X with a metric d.In topological dynamics one studies the asymptotic properties of continuous maps,such as topological entropy,topological pressure,chaos and Lyapunov exponents.Recur-rence property is an important issue and it has deep connections with the number the-ory,fractal geometry and differential equations.We focus on quantitative recurrence properties.In fact,we use the topological entropy,topological pressure,Hausdorff dimension to describe the recurrence properties or shrinking target problems.The thesis is devoted to study the set in the metric Diophantine approximation,via orbit-gluing properties in dynamical systems.In the viewpoint of multifractal analysis,we study some classes of level sets which describe the recurrence or shrink-ing behaviour in the metric Diophantine approximation.In Chapter 1,Boshernitzan's results are generalized to semigroup action dynamical systems.In Chapter 2,we in-vestigate quantitative recurrence properties for a class subshift with non-uniform and give an estimate of Hausdorff dimension of the level set of recurrence behaviour;In Chapter 3,study the topological pressure of the level set of saturated set of sys-tems with non-uniform structure and establish the conditional variational principle;In Chapter 4,we define a new level set and study the recurrence properties in the his-toric set;In Chapter 5,we define a new level set with some special shadowing time study the topological entropy of the set.This can be see a class of shrinking level set for non-symblic systems;In Chapter 6,we show that the non-dense orbit set has full topological pressure for topological dynamical systems.