Rainbow meanders and Cartesian billiards

Fiedler, Bernold; Castañeda, Pablo

doi:10.11606/issn.2316-9028.v6i2p247-275

Cited by 5 publications

(14 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar bending properties, quite curiously, were studied by Bonatti et al, and later by Vago, in the context of hyperbolic diffeomorphisms of a single-rectangle Markov partition; see [Bonatti et al, 1998;Vago, 2001]. For relations of closed meanders to plane Cartesian billiards see [Fiedler & Castañeda, 2012].…”

Section: Motivationmentioning

confidence: 74%

Reversibility, Continued Fractions, and Infinite Meander Permutations of Planar Homoclinic Orbits in Linear Hyperbolic Anosov Maps

Fiedler

2014

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

Meander permutations have been encountered in the context of Gauss words, singularity theory, Sturm global attractors, plane Cartesian billiards, and Temperley–Lieb algebras, among others. In this spirit, we attempt to investigate the difference of orderings of homoclinic orbits on the stable and unstable manifolds of a planar saddle. As an example, we consider reversible linear Anosov maps on the 2-torus, and their relation to continued fraction expansions.

show abstract

Section: Motivationmentioning

confidence: 74%

Reversibility, Continued Fractions, and Infinite Meander Permutations of Planar Homoclinic Orbits in Linear Hyperbolic Anosov Maps

Fiedler

2014

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

show abstract

“…See [DGG97] for further background on this correspondence. In [FC12] the close relation of Cartesian billiards and meanders has been studied. If the boundary of the billiard region is a single curve without self intersections (or, more generally, of self intersection only at integer lattice points -removable by making the corners of the boundary polygon round) then the billiard trajectories correspond to meander curves.…”

Section: As An Element Of a Temperley-lieb Algebramentioning

confidence: 99%

“…has no closed curve consisting of only one upper and one lower arc. See [FC12] for a complete proof.…”

Section: As a Cartesian Billiardmentioning

confidence: 99%

Connected components of meanders: Ⅰ. bi-rainbow meanders

Karnauhova¹,

Liebscher²

2017

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

Closed meanders are planar configurations of one or several disjoint closed Jordan curves intersecting a given line or curve transversely. They arise as shooting curves of parabolic PDEs in one space dimension, as trajectories of Cartesian billiards, and as representations of elements of Temperley-Lieb algebras.Given the configuration of intersections, for example as a permutation or an arc collection, the number of Jordan curves is unknown and needs to be determined.We address this question in the special case of bi-rainbow meanders, which are given as non-branched families (rainbows) of nested arcs. Easily obtainable results for small bi-rainbow meanders containing up to four families suggest an expression of the number of curves by the greatest common divisor (gcd) of polynomials in the sizes of the rainbow families.We prove however, that this is not the case. In fact, the number of connected components of bi-rainbow meanders with more than four families cannot be expressed as the gcd of polynomials in the sizes of the rainbows.On the other hand, we provide a complexity analysis of nose-retraction algorithms. They determine the number of connected components of arbitrary birainbow meanders in logarithmic time. In fact, the nose-retraction algorithms resemble the Euclidean algorithm, which is used to determine the gcd, in structure and complexity.Looking for a closed formula of the number of connected components, the noseretraction algorithm is as good as a gcd-formula and therefore as good as we can possibly expect. Meander curves

show abstract

“…By results of Zelenyak and Matano, the PDE (2.1) then possesses a gradient-like Morse structure due to a decreasing Lyapunov, or energy, or Morse, functional V . Throughout we assume all (9,2); see [14].…”

Section: Sturm Global Attractorsmentioning

confidence: 99%

“…We call A f a Sturm global attractor, to emphasize the particular origin from the one-dimensional parabolic PDE class (2.1). Information on the vertex pairs (v − , v + ) which do possess a heteroclinic edge (2.4), and on the pairs which don't, can be encoded in the (directed, acyclic, loop-free) connection graph C f ; see [14] and Fig. 2.1(a).…”

Section: Sturm Global Attractorsmentioning

confidence: 99%

Global Dynamics, Blow-Up, and Bianchi Cosmology

Ben-Gal,

Brehm,

Buchner

et al. 2016

Preprint

Self Cite

View full text Add to dashboard Cite

Many central problems in geometry, topology, and mathematical physics lead to questions concerning the long-time dynamics of solutions to ordinary and partial differential equations. Examples range from the Einstein field equations of general relativity to quasilinear reaction-advection-diffusion equations of parabolic type. Specific questions concern the convergence to equilibria, the existence of periodic, homoclinic, and heteroclinic solutions, and the existence and geometric structure of global attractors. On the other hand, many solutions develop singularities in finite time. The singularities have to be analyzed in detail before attempting to extend solutions beyond their singularities, or to understand their geometry in conjunction with globally bounded solutions. In this context we have also aimed at global qualitative descriptions of blow-up and grow-up phenomena.We survey some of our results supported in the framework of the Sonderforschungsbereich 647 "Space -Time -Matter" of the Deutsche Forschungsgemeinschaft.

show abstract

Rainbow meanders and Cartesian billiards

Cited by 5 publications

References 20 publications

Reversibility, Continued Fractions, and Infinite Meander Permutations of Planar Homoclinic Orbits in Linear Hyperbolic Anosov Maps

Reversibility, Continued Fractions, and Infinite Meander Permutations of Planar Homoclinic Orbits in Linear Hyperbolic Anosov Maps

Connected components of meanders: Ⅰ. bi-rainbow meanders

Global Dynamics, Blow-Up, and Bianchi Cosmology

Contact Info

Product

Resources

About