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Journal of Modern Dynamics

2014 , Volume 8 , Issue 2

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Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps
Bryce Weaver
2014, 8(2): 139-176 doi: 10.3934/jmd.2014.8.139 +[Abstract](397) +[PDF](419.8KB)
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.
Limit theorems for skew translations
Jory Griffin and Jens Marklof
2014, 8(2): 177-189 doi: 10.3934/jmd.2014.8.177 +[Abstract](422) +[PDF](237.1KB)
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.
On the singular-hyperbolicity of star flows
Yi Shi, Shaobo Gan and Lan Wen
2014, 8(2): 191-219 doi: 10.3934/jmd.2014.8.191 +[Abstract](413) +[PDF](280.5KB)
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.
Pseudo-automorphisms with no invariant foliation
Eric Bedford, Serge Cantat and Kyounghee Kim
2014, 8(2): 221-250 doi: 10.3934/jmd.2014.8.221 +[Abstract](391) +[PDF](315.9KB)
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.
Every action of a nonamenable group is the factor of a small action
Brandon Seward
2014, 8(2): 251-270 doi: 10.3934/jmd.2014.8.251 +[Abstract](372) +[PDF](228.9KB)
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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