Dimensional characteristics of the non-wandering sets of open billiards

Paul Wright

    Research output: ThesisDoctoral Thesis

    Abstract

    [Truncated] An open billiard is a dynamical system in which a pointlike particle moves at constant speed in an unbounded domain, reflecting off a boundary according to the classical laws of optics. This thesis is an investigation of dimensional characteristics of the non-wandering set of an open billiard in the exterior of three or more strictly convex bodies satisfying Ikawa’s no-eclipse condition. The billiard map for these systems is an axiom A diffeomorphism with a finite Markov partition. The non-wandering set is a hyperbolic set with stable and unstable manifolds satisfying a certain reflection property. The characteristics we investigate include the topological and measure-theoretic entropy, topological pressure, Lyapunov exponents, lower and upper box dimension and the Hausdorff dimension of the non-wandering set. In particular, we investigate the dependence of Hausdorff dimension on deformations to the boundary of the billiard obstacles. While the dependence of dimensional characteristics on perturbations of a system has been studied before, this is the first time this question has been answered for dynamical billiards.

    We find upper and lower bounds for the Hausdorff dimension using two different methods: one involving bounding the size of curves on convex fronts and the other using Bowen’s equation and the variational principle for topological pressure. Both methods lead to the same upper and lower bounds. In the first method, we use a well known recurrence relation for the successive curvatures of convex fronts to find bounds on the size of the fronts. This allows us to construct Lipschitz (but not bi-Lipszhitz) homeomorphisms between the non-wandering set and the one-sided symbol space. From there we obtain estimates of the dimension. This method has been previously used for open billiards in the plane. We extend it to higher dimensions and make improvements to the results in the plane.

    The second method is a more general approach from the dimension theory of dynamical systems. In the plane, the billiard map is conformal, meaning that its derivative is a multiple of an isometry. For conformal maps, the Hausdorff dimension of non-wandering sets is well-understood and satisfies Bowen’s equation. In higher dimensions, the billiard map is not conformal and the dimension only satisfies some estimates.

    LanguageEnglish
    QualificationDoctor of Philosophy
    StateUnpublished - 2014

    Fingerprint

    Billiards
    Hausdorff Dimension
    Topological Pressure
    Higher Dimensions
    Upper and Lower Bounds
    Dynamical system
    Markov Partition
    Box Dimension
    Hyperbolic Set
    Dimension Theory
    Axiom A
    Conformal Map
    Stable and Unstable Manifolds
    Strictly Convex
    Diffeomorphism
    Convex Body
    Unbounded Domain
    Recurrence relation
    Isometry
    Variational Principle

    Cite this

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    title = "Dimensional characteristics of the non-wandering sets of open billiards",
    abstract = "[Truncated] An open billiard is a dynamical system in which a pointlike particle moves at constant speed in an unbounded domain, reflecting off a boundary according to the classical laws of optics. This thesis is an investigation of dimensional characteristics of the non-wandering set of an open billiard in the exterior of three or more strictly convex bodies satisfying Ikawa’s no-eclipse condition. The billiard map for these systems is an axiom A diffeomorphism with a finite Markov partition. The non-wandering set is a hyperbolic set with stable and unstable manifolds satisfying a certain reflection property. The characteristics we investigate include the topological and measure-theoretic entropy, topological pressure, Lyapunov exponents, lower and upper box dimension and the Hausdorff dimension of the non-wandering set. In particular, we investigate the dependence of Hausdorff dimension on deformations to the boundary of the billiard obstacles. While the dependence of dimensional characteristics on perturbations of a system has been studied before, this is the first time this question has been answered for dynamical billiards. We find upper and lower bounds for the Hausdorff dimension using two different methods: one involving bounding the size of curves on convex fronts and the other using Bowen’s equation and the variational principle for topological pressure. Both methods lead to the same upper and lower bounds. In the first method, we use a well known recurrence relation for the successive curvatures of convex fronts to find bounds on the size of the fronts. This allows us to construct Lipschitz (but not bi-Lipszhitz) homeomorphisms between the non-wandering set and the one-sided symbol space. From there we obtain estimates of the dimension. This method has been previously used for open billiards in the plane. We extend it to higher dimensions and make improvements to the results in the plane. The second method is a more general approach from the dimension theory of dynamical systems. In the plane, the billiard map is conformal, meaning that its derivative is a multiple of an isometry. For conformal maps, the Hausdorff dimension of non-wandering sets is well-understood and satisfies Bowen’s equation. In higher dimensions, the billiard map is not conformal and the dimension only satisfies some estimates.",
    keywords = "Open billiard, No-eclipse condition, Non-wandering set, Dimension theory, Symbolic dynamics, Bowen's equation, Average conformal, Hausdorff dimension",
    author = "Paul Wright",
    year = "2014",
    language = "English",

    }

    Dimensional characteristics of the non-wandering sets of open billiards. / Wright, Paul.

    2014.

    Research output: ThesisDoctoral Thesis

    TY - THES

    T1 - Dimensional characteristics of the non-wandering sets of open billiards

    AU - Wright,Paul

    PY - 2014

    Y1 - 2014

    N2 - [Truncated] An open billiard is a dynamical system in which a pointlike particle moves at constant speed in an unbounded domain, reflecting off a boundary according to the classical laws of optics. This thesis is an investigation of dimensional characteristics of the non-wandering set of an open billiard in the exterior of three or more strictly convex bodies satisfying Ikawa’s no-eclipse condition. The billiard map for these systems is an axiom A diffeomorphism with a finite Markov partition. The non-wandering set is a hyperbolic set with stable and unstable manifolds satisfying a certain reflection property. The characteristics we investigate include the topological and measure-theoretic entropy, topological pressure, Lyapunov exponents, lower and upper box dimension and the Hausdorff dimension of the non-wandering set. In particular, we investigate the dependence of Hausdorff dimension on deformations to the boundary of the billiard obstacles. While the dependence of dimensional characteristics on perturbations of a system has been studied before, this is the first time this question has been answered for dynamical billiards. We find upper and lower bounds for the Hausdorff dimension using two different methods: one involving bounding the size of curves on convex fronts and the other using Bowen’s equation and the variational principle for topological pressure. Both methods lead to the same upper and lower bounds. In the first method, we use a well known recurrence relation for the successive curvatures of convex fronts to find bounds on the size of the fronts. This allows us to construct Lipschitz (but not bi-Lipszhitz) homeomorphisms between the non-wandering set and the one-sided symbol space. From there we obtain estimates of the dimension. This method has been previously used for open billiards in the plane. We extend it to higher dimensions and make improvements to the results in the plane. The second method is a more general approach from the dimension theory of dynamical systems. In the plane, the billiard map is conformal, meaning that its derivative is a multiple of an isometry. For conformal maps, the Hausdorff dimension of non-wandering sets is well-understood and satisfies Bowen’s equation. In higher dimensions, the billiard map is not conformal and the dimension only satisfies some estimates.

    AB - [Truncated] An open billiard is a dynamical system in which a pointlike particle moves at constant speed in an unbounded domain, reflecting off a boundary according to the classical laws of optics. This thesis is an investigation of dimensional characteristics of the non-wandering set of an open billiard in the exterior of three or more strictly convex bodies satisfying Ikawa’s no-eclipse condition. The billiard map for these systems is an axiom A diffeomorphism with a finite Markov partition. The non-wandering set is a hyperbolic set with stable and unstable manifolds satisfying a certain reflection property. The characteristics we investigate include the topological and measure-theoretic entropy, topological pressure, Lyapunov exponents, lower and upper box dimension and the Hausdorff dimension of the non-wandering set. In particular, we investigate the dependence of Hausdorff dimension on deformations to the boundary of the billiard obstacles. While the dependence of dimensional characteristics on perturbations of a system has been studied before, this is the first time this question has been answered for dynamical billiards. We find upper and lower bounds for the Hausdorff dimension using two different methods: one involving bounding the size of curves on convex fronts and the other using Bowen’s equation and the variational principle for topological pressure. Both methods lead to the same upper and lower bounds. In the first method, we use a well known recurrence relation for the successive curvatures of convex fronts to find bounds on the size of the fronts. This allows us to construct Lipschitz (but not bi-Lipszhitz) homeomorphisms between the non-wandering set and the one-sided symbol space. From there we obtain estimates of the dimension. This method has been previously used for open billiards in the plane. We extend it to higher dimensions and make improvements to the results in the plane. The second method is a more general approach from the dimension theory of dynamical systems. In the plane, the billiard map is conformal, meaning that its derivative is a multiple of an isometry. For conformal maps, the Hausdorff dimension of non-wandering sets is well-understood and satisfies Bowen’s equation. In higher dimensions, the billiard map is not conformal and the dimension only satisfies some estimates.

    KW - Open billiard

    KW - No-eclipse condition

    KW - Non-wandering set

    KW - Dimension theory

    KW - Symbolic dynamics

    KW - Bowen's equation

    KW - Average conformal

    KW - Hausdorff dimension

    M3 - Doctoral Thesis

    ER -