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Journal of Modern Dynamics

2011 , Volume 5 , Issue 1

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Shimura and Teichmüller curves
Martin Möller
2011, 5(1): 1-32 doi: 10.3934/jmd.2011.5.1 +[Abstract](1712) +[PDF](645.9KB)
We classify curves in the moduli space of curves $M_g$ that are both Shimura and Teichmüller curves: for both $g=3$ and $g=4$ there exists precisely one such curve, for $g=2$ and $g \geq 6$ there are no such curves.
   We start with a Hodge-theoretic description of Shimura curves and of Teichmüller curves that reveals similarities and differences of the two classes of curves. The proof of the classification relies on the geometry of square-tiled coverings and on estimating the numerical invariants of these particular fibered surfaces.
   Finally, we translate our main result into a classification of Teichmüller curves with totally degenerate Lyapunov spectrum.
Perfect retroreflectors and billiard dynamics
Pavel Bachurin, Konstantin Khanin, Jens Marklof and Alexander Plakhov
2011, 5(1): 33-48 doi: 10.3934/jmd.2011.5.33 +[Abstract](342) +[PDF](266.6KB)
We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.
Boundary unitary representations-irreducibility and rigidity
Uri Bader and Roman Muchnik
2011, 5(1): 49-69 doi: 10.3934/jmd.2011.5.49 +[Abstract](511) +[PDF](270.7KB)
Let $M$ be compact negatively curved manifold, $\Gamma =\pi_1(M)$ and $M$ be its universal cover. Denote by $B =\partial M$ the geodesic boundary of $M$ and by $\nu$ the Patterson-Sullivan measure on $X$. In this note we prove that the associated unitary representation of $\Gamma$ on $L^2(B,\nu)$ is irreducible. We also establish a new rigidity phenomenon: we show that some of the geometry of $M$, namely its marked length spectrum, is reflected in this $L^2$-representations.
Counting closed geodesics in moduli space
Alex Eskin and Maryam Mirzakhani
2011, 5(1): 71-105 doi: 10.3934/jmd.2011.5.71 +[Abstract](802) +[PDF](396.8KB)
We compute the asymptotics, as $R$ tends to infinity, of the number $N(R)$ of closed geodesics of length at most $R$ in the moduli space of compact Riemann surfaces of genus $g$. In fact, $N(R)$ is the number of conjugacy classes of pseudo-Anosov elements of the mapping class group of a compact surface of genus $g$ of translation length at most $R$.
Integrability and Lyapunov exponents
Andy Hammerlindl
2011, 5(1): 107-122 doi: 10.3934/jmd.2011.5.107 +[Abstract](396) +[PDF](218.5KB)
A smooth distribution, invariant under a dynamical system, integrates to give an invariant foliation, unless certain resonance conditions are present.
The Khinchin Theorem for interval-exchange transformations
Luca Marchese
2011, 5(1): 123-183 doi: 10.3934/jmd.2011.5.123 +[Abstract](407) +[PDF](590.3KB)
We define a Diophantine condition for interval-exchange transformations. When the number of intervals is two, that is, for rotations on the circle, our condition coincides with the classical Khinchin condition. We prove for interval-exchange transformations the same dichotomy as in the Khinchin Theorem. We also develop several results relating the Rauzy-Veech algorithm with homogeneous approximations for interval-exchange transformations.
Tori with hyperbolic dynamics in 3-manifolds
Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz and Raúl Ures
2011, 5(1): 185-202 doi: 10.3934/jmd.2011.5.185 +[Abstract](471) +[PDF](262.9KB)
Let $M$ be a closed orientable irreducible $3$-dimensional manifold. An embedded $2$-torus $\mathbb{T}$ is an Anosov torus if there exists a diffeomorphism $f$ over $M$ for which $\T$ is $f$-invariant and $f_\#|_\mathbb{T}:\pi_1(\mathbb{T})\to \pi_1(\mathbb{T})$ is hyperbolic. We prove that only few irreducible $3$-manifolds admit Anosov tori: (1) the $3$-torus $\mathbb{T}^3$; (2) the mapping torus of $-\Id$; and (3) the mapping tori of hyperbolic automorphisms of $\mathbb{T}^2$.
   This has consequences for instance in the context of partially hyperbolic dynamics of $3$-manifolds: if there is an invariant foliation $\mathcal{F}^{cu}$ tangent to the center-unstable bundle $E^c\oplus E^u$, then $\mathcal{F}^{cu}$ has no compact leaves [21]. This has led to the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle [21].

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