Ergodicity of toral linked twist mappings

Przytycki, Feliks

doi:10.24033/asens.1451

Cited by 32 publications

(38 citation statements)

References 5 publications

(7 reference statements)

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“…Both [11] and [37] in fact establish the Bernoulli property for the action of Φ on D 0 under conditions (i), (ii). We expect that these properties hold under weaker conditions, e.g., for smaller values of C 0 .…”

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confidence: 95%

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Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)

2004

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For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the distribution of eigenphases. If the map has one ergodic component, and is periodic on the remaining domains, we prove the Schnirelman-Zelditch-Colin de Verdière Theorem on the equidistribution of eigenfunctions with respect to the ergodic component of the classical map (quantum ergodicity). We apply our main theorems to quantised linked twist maps on the torus. In the Appendix, S. Zelditch connects these studies to some earlier results on 'pimpled spheres' in the setting of Riemannian manifolds. The common feature is a divided phase space with a periodic component.

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confidence: 95%

“…Ergodic properties of linked twist maps. The ergodic properties of linked twist maps are well understood [11,37]. Let [a i , b i ] (i = 1, 2) be subintervals of R/Z, and choose functions f i : R/Z → R with…”

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confidence: 99%

Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)

2004

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“…The class of maps we'll examine are toral linked twist maps. Linked twist maps (or LTMs) have been studied extensively [18][19][20][21][22][23][24][25]. They are non-uniformly hyperbolic, and are especially relevant here because of an intimate connection to the three-rod stirring protocols above.…”

Section: Toral Linked Twist Mapmentioning

confidence: 99%

“…This condition ensures that matrix L is hyperbolic and, consequently, that the toral LTM has an ergodic partition [28]. (With a slightly stronger condition, the toral LTM will also be Bernoulli [19].) To show convergence in this case, we transform the continued fraction into the form…”

Section: A Secondary Folding In Toral Ltmsmentioning

confidence: 99%

Topological Entropy and Secondary Folding

Tumasz

Thiffeault

2012

J Nonlinear Sci

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A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod-stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.

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“…Furthermore an application of the work of Katok et al [3] shows that the torus decomposes into countably many ergodic components, on each of which the restriction of g is either a Bernoulli map or the cross product of a Bernoulli map with a rotation. Przytycki has shown that if in addition for almost all points p, q ∈ T 2 and every pair of integers m, n large enough g m (l u (p)) ∩ g −n (l s (q)) = ∅ (where l u (p) is the local unstable manifold of p and l s (q) is the local unstable manifold of q), then g is a Bernoulli map [8]. The ergodicity of such a tltm may also be obtained from the more recent techniques described by Liverani and Wojtkowski [4].…”

Section: Introductionmentioning

confidence: 99%

A Bernoulli toral linked twist map without positive Lyapunov exponents

Nicol

1996

Proc. Amer. Math. Soc.

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Abstract. The presence of positive Lyapunov exponents in a dynamical system is often taken to be equivalent to the chaotic behavior of that system. We construct a Bernoulli toral linked twist map which has positive Lyapunov exponents and local stable and unstable manifolds defined only on a set of measure zero. This is a deterministic dynamical system with the strongest stochastic property, yet it has positive Lyapunov exponents only on a set of measure zero. In fact we show that for any map g in a certain class of piecewise linear Bernoulli toral linked twist maps, given any > 0 there is a Bernoulli toral linked twist map g with positive Lyapunov exponents defined only on a set of measure zero such that g is within of g in thed metric.

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Ergodicity of toral linked twist mappings

Cited by 32 publications

References 5 publications

Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)

Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)

Topological Entropy and Secondary Folding

A Bernoulli toral linked twist map without positive Lyapunov exponents

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