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Register a new userHausdorff dimension of three-period orbits in Birkhoff billiards
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We prove that the Hausdorff dimension of the set of three-period orbits in classical billiards is at most one. Moreover, if the set of three-period orbits has Hausdorff dimension one, then it has a tangent line at almost every point.
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An approach due to Wojtkovski [9], based on the Jacobi fields, is applied to study sets of 3-period orbits in billiards on hyperbolic plane and on two-dimensional sphere. It is found that the set of 3-period orbits in billiards on hyperbolic plane, as in the planar case, has zero measure. For the sphere, a new proof of Baryshnikovs theorem is obtained which states that 3-period orbits can form a set of positive measure provided a natural condition on the orbit length is satisfied.
In this paper we study the Birkhoff Normal Form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist coefficients in terms of the geometric parameters of the billiard table. As an application, we obtain characterizations of the nonlinear stability and local analytic integrability of the billiards around the elliptic periodic points.
As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension 3, we introduce a $C^2$-open set of diffeomorphisms of whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. Our proof employs the thicknesses of Cantor sets.
We deal with Besicovitchs problem of existence of discrete orbits for transitive cylindrical transformations $T_varphi:(x,t)mapsto(x+alpha,t+varphi(x))$ where $Tx=x+alpha$ is an irrational rotation on the circle $T$ and $varphi:TtoR$ is continuous, i.e. we try to estimate how big can be the set $D(alpha,varphi):={xinT:|varphi^{(n)}(x)|to+inftytext{as}|n|to+infty}$. We show that for almost every $alpha$ there exists $varphi$ such that the Hausdorff dimension of $D(alpha,varphi)$ is at least $1/2$. We also provide a Diophantine condition on $alpha$ that guarantees the existence of $varphi$ such that the dimension of $D(alpha,varphi)$ is positive. Finally, for some multidimensional rotations $T$ on $T^d$, $dgeq3$, we construct smooth $varphi$ so that the Hausdorff dimension of $D(alpha,varphi)$ is positive.
In this paper, we study the Hausdorff and the box dimensions of closed invariant subsets of the space of pointed trees, equipped with a pseudogroup action. This pseudogroup dynamical system can be regarded as a generalization of a shift space. We show that the Hausdorff dimension of the space of pointed trees is infinite, and the union of closed invariant subsets with dense orbit and non-equal Hausdorff and box dimensions is dense in the space of pointed trees. We apply our results to the problem of embedding laminations into differentiable foliations of smooth manifolds. To admit such an embedding, a lamination must satisfy at least the following two conditions: first, it must admit a metric and a foliated atlas, such that the generators of the holonomy pseudogroup, associated to the atlas, are bi-Lipschitz maps relative to the metric. Second, it must admit an embedding into a manifold, which is a bi-Lipschitz map. A suspension of the pseudogroup action on the space of pointed graphs gives an example of a lamination where the first condition is satisfied, and the second one is not satisfied, with Hausdorff dimension of the space of pointed trees being the obstruction to the existence of a bi-Lipschitz embedding.
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