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Journal of Modern Dynamics

2012 , Volume 6 , Issue 4

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Weak mixing suspension flows over shifts of finite type are universal
Anthony Quas and Terry Soo
2012, 6(4): 427-449 doi: 10.3934/jmd.2012.6.427 +[Abstract](312) +[PDF](547.7KB)
Abstract:
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 [5] and Ratner [31], 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.
An algebraic characterization of expanding Thurston maps
Peter Haïssinsky and Kevin M. Pilgrim
2012, 6(4): 451-476 doi: 10.3934/jmd.2012.6.451 +[Abstract](283) +[PDF](301.5KB)
Abstract:
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 $g$.
Ergodic infinite group extensions of geodesic flows on translation surfaces
David Ralston and Serge Troubetzkoy
2012, 6(4): 477-497 doi: 10.3934/jmd.2012.6.477 +[Abstract](297) +[PDF](215.8KB)
Abstract:
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.
Connecting orbits for families of Tonelli Hamiltonians
Vito Mandorino
2012, 6(4): 499-538 doi: 10.3934/jmd.2012.6.499 +[Abstract](316) +[PDF](336.7KB)
Abstract:
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 [20] and Le Calvez [15]. Our approach is based on weak KAM theory and is close to the one used by Bernard in [3] to study the case of a single Tonelli Hamiltonian.
Quadratic irrationals and linking numbers of modular knots
Dubi Kelmer
2012, 6(4): 539-561 doi: 10.3934/jmd.2012.6.539 +[Abstract](352) +[PDF](241.8KB)
Abstract:
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.
A dynamical approach to Maass cusp forms
Anke D. Pohl
2012, 6(4): 563-596 doi: 10.3934/jmd.2012.6.563 +[Abstract](264) +[PDF](778.9KB)
Abstract:
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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