We show that a topologically mixing $C^infty$ Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential.
We classify five dimensional Anosov flows with smooth decomposition which are in addition transversely symplectic. Up to finite covers and a special time change, we find exectly the suspensions of symplectic hyperbolic automorphisms of four dimension
al toris, and the geodesic flows of three dimensional hyperbolic manifolds.
We classify quasiconformal Anosov flows whose strong stable and unstable distributions are at least two dimensional and the sum of these two distributions is smooth. We deduce from this classification result the complete classification of volume-pres
erving quasiconformal diffeomorphisms whose stable and unstable distributions are at least two dimensional. Our central idea is to take a good time change so that perodic orbits are equi-distributed with respect to a lebesgue measure.
We study the cohomological pressure introduced by R.Sharp (defined by using topological pressures of certain potentials of Anosov flows). In particular, we get the rigidity in the case that this pressure coincides with the metrical entropy, generalis
ing related rigidity results of A.Katok and P. Foulon.
We prove that any strongly mixing action of a countable abelian group on a probability space has higher order mixing properties. This is achieved via introducing and utilizing $mathcal R$-limits, a notion of convergence which is based on the classica
l Ramsey Theorem. $mathcal R$-limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of uniform density one and IP$^*$. While the main goal of this paper is to establish a $textit{universal}$ property of strongly mixing actions of countable abelian groups, our results, when applied to $mathbb Z$-actions, offer a new way of dealing with strongly mixing transformations. In particular, we obtain several new characterizations of strong mixing for $mathbb Z$-actions, including a result which can be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his proof of Szemeredis theorem. We also demonstrate the versatility of $mathcal R$-limits by obtaining new characterizations of higher order weak and mild mixing for actions of countable abelian groups.
For the time-one map $f$ of a contact Anosov flow on a compact Riemann manifold $M$, satisfying a certain regularity condition, we show that given a Gibbs measure on $M$, a sufficiently large Pesin regular set $P_0$ and an arbitrary $delta in (0,1)$,
there exist positive constants $C$ and $c$ such that for any integer $n geq 1$, the measure of the set of those $xin M$ with $f^k(x) otin P_0$ for at least $delta n$ values of $k = 0,1, ldots,n-1$ does not exceed $C e^{-cn}$.