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Journal of Modern Dynamics

2013 , Volume 7 , Issue 1

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Divergent trajectories in the periodic wind-tree model
Vincent Delecroix
2013, 7(1): 1-29 doi: 10.3934/jmd.2013.7.1 +[Abstract](105) +[PDF](419.8KB)
Abstract:
The periodic wind-tree model is a family $T(a,b)$ of billiards in the plane in which identical rectangular scatterers of size $a \times b$ are disposed periodically at each integer point. In that model, the recurrence is generic with respect to the parameters $a$, $b$, and the angle $\theta$ of initial direction of the particule. In contrast, we prove that for some parameters $(a,b)$ the set of angles $\theta$ for which the billiard flow is divergent has Hausdorff dimension greater than one half.
Growth of periodic orbits and generalized diagonals for typical triangular billiards
Dmitri Scheglov
2013, 7(1): 31-44 doi: 10.3934/jmd.2013.7.31 +[Abstract](92) +[PDF](263.5KB)
Abstract:
We prove that for any $\epsilon>0$ the growth rate $P_n$ of generalized diagonals or periodic orbits of a typical (in the Lebesgue measure sense) triangular billiard satisfies: $P_n < Ce^{n^{\sqrt{3}-1+\epsilon}}$. This provides an explicit subexponential estimate on the triangular billiard complexity and answers a long-standing open question for typical triangles. This also makes progress towards a solution of Problem 3 in Katok's list of "Five most resistant problems in dynamics". The proof uses essentially new geometric ideas and does not rely on the rational approximations.
On bounded cocycles of isometries over minimal dynamics
Daniel Coronel, Andrés Navas and Mario Ponce
2013, 7(1): 45-74 doi: 10.3934/jmd.2013.7.45 +[Abstract](96) +[PDF](282.8KB)
Abstract:
We show the following geometric generalization of a classical theorem of W. H. Gottschalk and G. A. Hedlund: a skew action induced by a cocycle of (affine) isometries of a Hilbert space over a minimal dynamical system has a continuous invariant section if and only if the cocycle is bounded. Equivalently, the associated twisted cohomological equation has a continuous solution if and only if the cocycle is bounded. We interpret this as a version of the Bruhat-Tits Center Lemma in the space of continuous functions. Our result also holds when the fiber is a proper CAT(0) space. One of the applications concerns matrix cocycles. Using the action of $\mathrm{GL} (n,\mathbb{R})$ on the (nonpositively curved) space of positively definite matrices, we show that every bounded linear cocycle over a minimal dynamical system is cohomologous to a cocycle taking values in the orthogonal group.
The Cayley-Oguiso automorphism of positive entropy on a K3 surface
Dino Festi, Alice Garbagnati, Bert Van Geemen and Ronald Van Luijk
2013, 7(1): 75-97 doi: 10.3934/jmd.2013.7.75 +[Abstract](91) +[PDF](662.2KB)
Abstract:
Recently Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy. The K3 surfaces used by Oguiso have a particular rank two Picard lattice. We show, using results of Beauville, that these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley constructed an automorphism of such determinantal surfaces. We show that Cayley's automorphism coincides with Oguiso's free automorphism. We also exhibit an explicit example of a determinantal quartic whose Picard lattice has exactly rank two and for which we thus have an explicit description of the automorphism.
Topological characterization of canonical Thurston obstructions
Nikita Selinger
2013, 7(1): 99-117 doi: 10.3934/jmd.2013.7.99 +[Abstract](81) +[PDF](214.4KB)
Abstract:
Let $f$ be an obstructed Thurston map with canonical obstruction $\Gamma_f$. We prove the following generalization of Pilgrim's conjecture: if the first-return map $F$ of a periodic component $C$ of the topological surface obtained from the sphere by pinching the curves of $\Gamma_f$ is a Thurston map then the canonical obstruction of $F$ is empty. Using this result, we give a complete topological characterization of canonical Thurston obstructions.
Remarks on quantum ergodicity
Gabriel Rivière
2013, 7(1): 119-133 doi: 10.3934/jmd.2013.7.119 +[Abstract](123) +[PDF](197.6KB)
Abstract:
We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in geometric situations in which the Liouville measure is not (or not known to be) ergodic.
Strata of abelian differentials and the Teichmüller dynamics
Dawei Chen
2013, 7(1): 135-152 doi: 10.3934/jmd.2013.7.135 +[Abstract](109) +[PDF](221.5KB)
Abstract:
This paper focuses on the interplay between the intersection theory and the Teichmüller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a nonsimple zero, and verify a number of extremal effective divisors on the moduli space of pointed curves in low genus.

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