Journal of Modern Dynamics
April 2014 , Volume 8 , Issue 2
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We obtain a precise asymptotic formula for the growth rate of periodic orbits of the geodesic flow over metrics on surfaces with negative curvature outside of a disjoint union of radially symmetric focusing caps of positive curvature. This extends results of G. Margulis and G. Knieper for negative and nonpositive curvature respectively.
Bufetov, Bufetov-Forni and Bufetov-Solomyak have recently proved limit theorems for translation flows, horocycle flows and tiling flows, respectively. We present here analogous results for skew translations of a torus.
We prove for a generic star vector field $X$ that if, for every chain recurrent class $C$ of $X$, all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular-hyperbolic. We also prove that every Lyapunov stable chain recurrent class of a generic star vector field is singular-hyperbolic. As a corollary, we prove that the chain recurrent set of a generic 4-dimensional star flow is singular-hyperbolic.
We construct an example of a birational transformation of a rational threefold for which the first and second dynamical degrees coincide and are $>1$, but which does not preserve any holomorphic (singular) foliation. In particular, this provides a negative answer to a question of Guedj. On our way, we develop several techniques to study foliations which are invariant under birational transformations.
It is well known that if $G$ is a countable amenable group and $G ↷ (Y, \nu)$ factors onto $G ↷ (X, \mu)$, then the entropy of the first action must be at least the entropy of the second action. In particular, if $G ↷ (X, \mu)$ has infinite entropy, then the action $G ↷ (Y, \nu)$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable nonamenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G ↷ (X, \mu)$ there is a $G$-invariant probability measure $\nu$ on $n^G$ such that $G ↷ (n^G, \nu)$ factors onto $G ↷ (X, \mu)$. For many nonamenable groups, $n$ can be chosen to be $4$ or smaller. We also obtain a similar result with respect to continuous actions on compact spaces and continuous factor maps.
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