scholarly journals Homotopical complexity of 2D billiard orbits

2011 ◽  
Vol 48 (4) ◽  
pp. 540-562
Author(s):  
Lee Goswick ◽  
Nándor Simányi
Keyword(s):  
Fundamental Group ◽  
Smooth Boundary ◽  
Natural Habitat ◽  
Hyperbolic Group ◽  
Height Function ◽  
Upper And Lower Bounds ◽  
Closed Loops ◽  
Rotation Numbers ◽  
Sinai Billiard ◽  
Rotation Set

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures live in the (commutative) vector space ℝn, if the toroidal billiard is given on the flatn-torus. The billiard trajectories, being curves, often getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e. the billiard flow on the 2-torus with a single, strictly convex obstacle (with smooth boundary) removed. This fundamental group is known to be the groupF2freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov’s sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classically expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits.The natural habitat for these objects is the infinite cone erected upon the Cantor set Ends (F2) of all ŋds" of the hyperbolic groupF2. An element of Ends (F2) describes the direction in (the Cayley graph of) the groupF2in which the considered trajectory escapes to infinity, whereas the height functiont(t≧ 0) of the cone gives us the average speed at which this escape takes place.The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}, and any directione∈ Ends (F2) for the escape is feasible with any prescribed speeds, 0 ≦s≦\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}/2. This means that the radial upper and lower bounds for the rotation setRare actually pretty close to each other.

scholarly journals Homotopical complexity of a billiard flow on the 3D flat torus with two cylindrical obstacles

2017 ◽  
Vol 39 (4) ◽  
pp. 1071-1081
Author(s):  
CALEB C. MOXLEY ◽  
NANDOR J. SIMANYI
Keyword(s):  
Natural Habitat ◽  
Hyperbolic Group ◽  
Height Function ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Flat Torus ◽  
Average Speed ◽  
Rotation Vectors ◽  
Rotation Set

We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two disjoint and orthogonal toroidal (cylindrical) scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(G)$ of all ‘ends’ of the hyperbolic group $G=\unicode[STIX]{x1D70B}_{1}(\mathbf{Q})$. An element of $\text{Ends}(G)$ describes the direction in (the Cayley graph of) the group $G$ in which the considered trajectory escapes to infinity, whereas the height function $s$ ($s\geq 0$) of the cone gives us the average speed at which this escape takes place. The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding $\sqrt{3}$ and, in any direction $e\in \text{Ends}(\unicode[STIX]{x1D70B}_{1}({\mathcal{Q}}))$, the escape is feasible with any prescribed speed $s$, $0\leq s\leq 1/(\sqrt{6}+2\sqrt{3})$. This means that the radial upper and lower bounds for the rotation set $R$ are actually pretty close to each other. Furthermore, we prove the convexity of the set $\mathit{AR}$ of constructible rotation vectors, and that the set of rotation vectors of periodic orbits is dense in $\mathit{AR}$. We also provide effective lower and upper bounds for the topological entropy of the studied billiard flow.

scholarly journals AN ELEMENTARY PROOF OF A THEOREM BY MATSUMOTO

2016 ◽  
Vol 227 ◽  
pp. 77-85 ◽  
Author(s):  
LUIS HERNÁNDEZ-CORBATO
Keyword(s):  
Elementary Proof ◽  
Alternative Proof ◽  
Rotation Numbers ◽  
Rotation Set ◽  
Prime End

Matsumoto proved in, [Prime end rotation numbers of invariant separating continua of annular homeomorphisms, Proc. Amer. Math. Soc. 140(3) (2012), 839–845.] that the prime end rotation numbers associated to an invariant annular continuum are contained in its rotation set. An alternative proof of this fact using only simple planar topology is presented.

scholarly journals Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

scholarly journals Surface groups within Baumslag doubles

2011 ◽  
Vol 54 (1) ◽  
pp. 91-97 ◽  
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Orientable Surface ◽  
Hyperbolic Surface ◽  
Surface Groups ◽  
Word Hyperbolic Group ◽  
Genus 2

AbstractA conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

scholarly journals ON THE REMAK HEIGHT, THE MAHLER MEASURE AND CONJUGATE SETS OF ALGEBRAIC NUMBERS LYING ON TWO CIRCLES

2001 ◽  
Vol 44 (1) ◽  
pp. 1-17 ◽  
Author(s):  
A. Dubickas ◽  
C. J. Smyth
Keyword(s):  
Lower Bounds ◽  
Height Function ◽  
Complete Characterization ◽  
Mahler Measure ◽  
The Other ◽  
Upper And Lower Bounds ◽  
Algebraic Numbers ◽  
Salem Numbers ◽  
Primary 11R06

AbstractWe define a new height function $\mathcal{R}(\alpha)$, the Remak height of an algebraic number $\alpha$. We give sharp upper and lower bounds for $\mathcal{R}(\alpha)$ in terms of the classical Mahler measure $M(\alpha)$. Study of when one of these bounds is exact leads us to consideration of conjugate sets of algebraic numbers of norm $\pm 1$ lying on two circles centred at 0. We give a complete characterization of such conjugate sets. They turn out to be of two types: one related to certain cubic algebraic numbers, and the other related to a non-integer generalization of Salem numbers which we call extended Salem numbers.AMS 2000 Mathematics subject classification: Primary 11R06

scholarly journals Hyperbolic groups with boundary an n-dimensional Sierpinski space

2019 ◽  
Vol 11 (01) ◽  
pp. 233-247
Author(s):  
Jean-François Lafont ◽  
Bena Tshishiku
Keyword(s):  
Fundamental Group ◽  
Negative Curvature ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Manifolds With Boundary ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Visual Boundary ◽  
Aspherical Manifolds ◽  
Torsion Free

For [Formula: see text], we show that if [Formula: see text] is a torsion-free hyperbolic group whose visual boundary [Formula: see text] is an [Formula: see text]-dimensional Sierpinski space, then [Formula: see text] for some aspherical [Formula: see text]-manifold [Formula: see text] with non-empty boundary. Concerning the converse, we construct, for each [Formula: see text], examples of aspherical manifolds with boundary, whose fundamental group [Formula: see text] is hyperbolic, but with visual boundary [Formula: see text] not homeomorphic to [Formula: see text]. Our examples even support (metric) negative curvature, and have totally geodesic boundary.

scholarly journals Rank and Nielsen equivalence in hyperbolic extensions

2019 ◽  
Vol 29 (04) ◽  
pp. 615-625
Author(s):  
Spencer Dowdall ◽  
Samuel J. Taylor
Keyword(s):  
Fundamental Group ◽  
Large Class ◽  
Free Groups ◽  
Hyperbolic Group ◽  
Group Extensions ◽  
Convex Cocompact

In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered [Formula: see text]-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of [Formula: see text].

Rotation and periodicity in plane separating continua

1991 ◽  
Vol 11 (4) ◽  
pp. 619-631 ◽  
Author(s):  
Marcy Barge ◽  
Richard M. Gillette
Keyword(s):  
Periodic Orbit ◽  
Convex Hull ◽  
Rotation Number ◽  
Closed Interval ◽  
Rotation Numbers ◽  
Indecomposable Continuum ◽  
Rotation Set

AbstractWe prove that ifFis an orientation-preserving homeomorphism of the plane that leaves invariant a continuum Λ which irreducibly separates the plane into exactly two domains, then the convex hull of the rotation set ofFrestricted to Λ is a closed interval and each reduced rational in this interval is the rotation number of a periodic orbit in Λ. We also show that the interior and exterior rotation numbers ofFassociated with Λ are contained in the convex hull of the rotation set ofFrestricted to Λ and that if this set is nondegenerate then Λ is an indecomposable continuum.

scholarly journals Hyperbolic Groups That Are Not Commensurably Co-Hopfian

2020 ◽  
Author(s):  
Emily Stark ◽  
Daniel J Woodhouse
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Torsion Free ◽  
Simple Surface

Abstract Sela proved that every torsion-free one-ended hyperbolic group is co-Hopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably co-Hopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably co-Hopfian.

scholarly journals On distribution densities of algebraic points under different height functions

2021 ◽  
Vol 65 (6) ◽  
pp. 647-653
Author(s):  
D. V. Koleda
Keyword(s):  
Distribution Density ◽  
Height Function ◽  
Upper And Lower Bounds ◽  
Random Polynomial ◽  
Algebraic Numbers ◽  
Distribution Of Zeros ◽  
Fixed Degree ◽  
Height Functions ◽  
Distribution Of Points ◽  
Conjugate Algebraic Numbers

In the article we consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed degree. The distribution is introduced using a height function. We have obtained universal upper and lower bounds of the distribution density of such points using an arbitrary height function. We have shown how from a given joint density function of coefficients of a random polynomial of degree n, one can construct such a height function H that the polynomials q of degree n uniformly chosen under H[q] ≤1 have the same distribution of zeros as the former random polynomial.

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