Journal of Modern Dynamics
April 2013 , Volume 7 , Issue 2
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We study the abundance of a special class of elliptic islands for the standard family of area-preserving diffeomorphism for large parameter values, i.e., far from the KAM regime. Outside a bounded set of parameter values, we prove that the measure of the set of parameter values for which an infinite number of such elliptic islands coexist is zero. On the other hand, we construct a positive Hausdorff dimension set of arbitrarily large parameter values for which the associated standard map admits infinitely many elliptic islands whose centers accumulate on a locally maximal hyperbolic set.
We define the Weierstrass filtration for Teichmüller curves and construct the Harder-Narasimhan filtration of the Hodge bundle of a Teichmüller curve in hyperelliptic loci and low-genus nonvarying strata. As a result we obtain the sum of Lyapunov exponents of Teichmüller curves in these strata.
We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of nonparabolic elements in the periodic field of certain translation surfaces.
We provide abstract conditions which imply the existence of a robustly invariant neighborhood of a global section of a fiber bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fiber is the space of jets (which are described by local manifolds). As a consequence we obtain sets of manifolds (e.g., approximations of stable manifolds) that are left invariant for all negative times by the flow and its small perturbations. Finally, we show that the latter result can be used to easily fix a mistake recently uncovered in the paper Smooth Anosov flows: correlation spectra and stability  by the present authors.
We show that every group $G$ with no cyclic subgroup of finite index that acts properly and cocompactly by isometries on a proper geodesic Gromov-hyperbolic space $X$ is growth-tight. In other words, the exponential growth rate of $G$ for the geometric (pseudo)-distance induced by $X$ is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result unifies and extends previous works of Arzhantseva-Lysenok and Sambusetti using a geometric approach.
We give effective bounds on the deviation of ergodic averages for the horocycle flow on the unit tangent bundle of a noncompact hyperbolic surface of finite area. The bounds depend on the small eigenvalues of the Laplacian and on the rate of excursion into cusps for the geodesic corresponding to the given initial point. We also prove Ω-results which show that in a certain sense our bounds are essentially the best possible for any given initial point.
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