Poincare Recurrence Of Two-dimensional Nonuniformly Hyperbolic Transformations

In this paper,we mainly study the quantitative behavior of Poincare recurrence of two-dimensional nonuniformly. The main content as follows:In chapter 2,we first study the quantitative behavior of Poincare recurrence theorem for a class of transformations on [0,1]~2, which includes transformations obtained by a Poincare section of the Lorenz equation. We show that the recurrence rate to each point coincide almost everywhere with the Hausdorff dimension of the measure with positive entropy. We develop the results about relationships between Poincare recurrence, Lyapunov exponent and entropy.In chapter 3,we study the quantitative behavior of Poincare recurrence theorem for Lauwerier transformation. We show that the upper-recurrence rate to each point coincide almost everywhere with the upper-pointwise dimension and the lower-recurrence rate to each point coincide almost everywhere with the lower-pointwise dimension. We develop the result about relationship between Poincare recurrence and pointwise dimension.