Ilya Shkredov

On quantitative reccurence

Let $X$ be a metric space with metric $d(\cdot ,\cdot )$ and a Borel sigma--algebra of measurable sets $\Phi$. Let $T$ be a measure preserving transformation of a measure space $(X,\Phi,\mu)$ and let us assume that measure of $X$ is finite. The well--known Poincare theorem asserts that for almost every point $x \in X $:
$$ \forall \eps > 0 ~\forall K > 0 ~\exists t > K : d(T^tx,x) < \eps .$$
We obtain some quantitative generalizations of this theorem for the case of totally bounded metric spaces and metric spaces with finite Hausdorff dimension.