Journal of Modern Dynamics
October 2012 , Volume 6 , Issue 4
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Let $S$ be an ergodic measure-preserving automorphism on a nonatomic probability space, and let $T$ be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Hölder ceiling function. We show that if the measure-theoretic entropy of $S$ is strictly less than the topological entropy of $T$, then there exists an embedding of the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen  and Ratner , we also obtain an embedding of the measure-preserving automorphism into a geodesic flow whenever the measure-theoretic entropy of $S$ is strictly less than the topological entropy of the time-one map of the geodesic flow.
Let $f\colon S^2 \to S^2$ be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for $f$ to be homotopic, relative to its postcritical set, to an expanding map $g$.
We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. K. Frączek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases.
We investigate the existence of Arnold diffusion-type orbits for systems obtained by iterating in any order the time-one maps of a family of Tonelli Hamiltonians. Such systems are known as 'polysystems' or 'iterated function systems'. When specialized to families of twist maps on the cylinder, our results are similar to those obtained by Moeckel  and Le Calvez . Our approach is based on weak KAM theory and is close to the one used by Bernard in  to study the case of a single Tonelli Hamiltonian.
A closed geodesic on the modular surface gives rise to a knot on the 3-sphere with a trefoil knot removed, and one can compute the linking number of such a knot with the trefoil knot. We show that, when ordered by their length, the set of closed geodesics having a prescribed linking number become equidistributed on average with respect to the Liouville measure. We show this by using the thermodynamic formalism to prove an equidistribution result for a corresponding set of quadratic irrationals on the unit interval.
For nonuniform cofinite Fuchsian groups $\Gamma$ that satisfy a certain additional geometric condition, we show that the Maass cusp forms for $\Gamma$ are isomorphic to $1$-eigenfunctions of a finite-term transfer operator. The isomorphism is constructive.
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