On the topological entropy of families of braids

Hall, Toby; Yurttaş, S. Öykü

doi:10.1016/j.topol.2009.01.005

Cited by 22 publications

(49 citation statements)

References 12 publications

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“…The dilatation of a given a pseudo-Anosov isotopy class equals the spectral radius of its Dynnikov matrix. Making use of this fact, in [20] we gave an alternative approach to compute the dilatation of each member of an infinite family of pseudo-Anosov isotopy classes on D n using Dynnikov matrices. The aim of this paper is to show that it is not only the dilatation but the whole set of eigenvalues (up to roots of unity) that train track transition and Dynnikov matrices share.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The next section describes the Dynnikov coordinate system which puts global coordinates on MF n [6,17,[20][21][22]].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Knowing from [16,20] that both Dynnikov matrices and train track transition matrices record the action of a pseudo-Anosov braid on the same space PMF n , we would like to know exactly what the new action tells us. Note that the dimensions of a Dynnikov and train track transition matrix for the same braid are in general different.…”

Section: Resultsmentioning

confidence: 99%

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Dynnikov and train track transition matrices of pseudo-Anosov braids

Yurttaş¹

2015

DCDS-A

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We compare the spectra of Dynnikov matrices with the spectra of the train track transition matrices of a given pseudo-Anosov braid on the finitely punctured disk, and show that these matrices are isospectral up to roots of unity and zeros under some particular conditions. It is shown, via examples, that Dynnikov matrices are much easier to compute than transition matrices, and so yield data that was previously inaccessible.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The next section describes the Dynnikov coordinate system which puts global coordinates on MF n [6,17,[20][21][22]].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Dynnikov and train track transition matrices of pseudo-Anosov braids

Yurttaş¹

2015

DCDS-A

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“…generalized Dynnikov coordinates of measured foliations [5]: the transverse measure on the foliation [4,7,8] assigns to each element in A k,n a nonnegative real number, and hence each measured foliation is described by an element of (R …”

Section: Remark 216 Generalized Dynnikov Coordinates For Integral Lamentioning

confidence: 99%

Integral laminations on nonorientable surfaces

Yurttaş¹,

Pamuk²

2018

Turk J Math

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“…40,45 The actual topological entropy produced by the flow map, h f , can be determined by computing the asymptotic stretching rate of topologically non-trivial lines, 46 such as lines that join a periodic point with the outer boundary. We estimate h f by following two orthogonal lines that are initially along the lines x ¼ a/2 and y ¼ 0, respectively, as they are stretched over 6-10 periods of the flow.…”

Section: Braiding Of Periodic Orbits In a Lid-driven Cavity Flowmentioning

confidence: 99%

Topological chaos, braiding and bifurcation of almost-cyclic sets

Grover

Ross

Stremler

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or "ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems. When a body of fluid moves, whether it be in the atmosphere, an ocean, or a kitchen sink, there are often regions of fluid that move together for an extended period of time. As these coherent sets of fluid trace out trajectories in space and time, they can be thought of as "stirring" the surrounding fluid. The entanglement of these trajectories as they "braid" around each other is connected to the level of chaos present in the fluid system. When the sets return periodically to their initial positions in a twodimensional, time-dependent system, the entanglement of their trajectories can be used to predict a lower bound on the rate of stretching in the surrounding domain. We examine the trajectories of such "almost-cyclic sets" in a lid-driven cavity flow and demonstrate that this combination of topological analysis with set-oriented methods can be an effective means of predicting chaos. The characterization of the entanglement and associated prediction of stretching are achieved through application of the Thurston-Nielsen Classification Theorem, which in general classifies the topological complexity of homeomorphisms of punctured surfaces. While a rigorous lower bound on topological entropy is not available in the absence of exactly periodic braiding structures, our approach finds the "topological skeleton" that can be used to get an approximate rate of stretching.

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On the topological entropy of families of braids

Cited by 22 publications

References 12 publications

Dynnikov and train track transition matrices of pseudo-Anosov braids

Dynnikov and train track transition matrices of pseudo-Anosov braids

Integral laminations on nonorientable surfaces

Topological chaos, braiding and bifurcation of almost-cyclic sets

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