Abstract: Smooth dynamical system can exhibit a wide range of behaviors rangingfrom most regular, such as periodic or quasiperiodic to most
stochastic such as Anosov diffeomorhphisms. Anosov maps enjoy a wide
range of statistical properties including ergodicity, weak mixing,
mixing, positive entropy, K property, Bernoulli property, Central
Limit Theorem, Large Deviations Bounds and Exponential Mix- ing. In
this talk we discuss the following problem:

Given two disjoint subsets P and A from the list above does there
exist a system in which all properties from P are present and all
properties from A are absent?

 After reviewing previous work on this topic we present several new
examples constructed in a joint work with Changguang Dong, Adam
Kanigowski and Peter Nandori.