Pseudo-Anosov homeomorphisms with quadratic expansion

Franks, John; Rykken, Elyn

doi:10.1090/s0002-9939-99-04731-0

Cited by 13 publications

(5 citation statements)

References 6 publications

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“…In [2], Arnoux and Fathi give an example of a pseudo-Anosov diffeomorphism f with orientable foliations on a surface of genus 3 whose dilatation coefficient is algebraic of degree 4 and such that f does not factor via a branched covering to a pseudo-Anosov map with irreducible action on its rational homology §. A similar example is impossible for λ that is a quadratic irrationality; indeed, we recover the result (Theorem 2.3) from [7]. Proof.…”

Section: Proof Of Corollary 3 From Theoremsupporting

confidence: 76%

Hyperbolic pseudo-Anosov maps almost everywhere embed into a toral automorphism

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Kwapisz

2006

Ergod. Th. Dynam. Sys.

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Fathi and Franks showed that a pseudo-Anosov diffeomorphism $f$ with orientable foliations and dilation coefficient $\lambda$ with no conjugates (over ${\mathbb Q}$) in the unit circle factors onto a (homologically non-trivial) invariant subset of a hyperbolic toral automorphism. After recounting this result, we show that the factor map is either almost everywhere one-to-one or almost everywhere $m$-to-one for some $m>1$ and the pseudo-Anosov map $f$ is an $m$-to-one ramified covering of another pseudo-Anosov (or Anosov) map on a surface of smaller genus. As a corollary, any pseudo-Anosov diffeomorphism with orientable foliations and hyperbolic action on the first homologies almost everywhere embeds into a hyperbolic toral automorphism.

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Section: Proof Of Corollary 3 From Theoremsupporting

confidence: 76%

Hyperbolic pseudo-Anosov maps almost everywhere embed into a toral automorphism

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Kwapisz

2006

Ergod. Th. Dynam. Sys.

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“…The existence of β 1 also follows from Franks and Rykken [FR99], who show that β 1 is always a branched cover with branch points and their images singularities. Thus β 1 is smooth at all but finitely many points and the preimages β 1 −1 (x) are finite sets with a uniformly bounded cardinality.…”

Section: Connections To Other Standard Measures and Distributionsmentioning

confidence: 71%

On eigen-structures for pseudoAnosov maps

Boyland

2010

Preprint

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We investigate various structures associated with the hyperbolic Markov and homological spectra of a pseudo-Anosov map φ on a surface. Each unstable eigenvalue of the action of φ on first cohomolgy yields an eigen-cocycle that is transverse and holonomy invariant to the stable foliation F s of φ. Each unstable eigenvalue µ of a Markov transition matrix for φ yields a holonomy invariant additive function G on transverse arcs to F s with φ * G = µG. Except when µ is the dilation of φ, these transverse arc functions do not yield measures, but rather holonomy invariant eigendistributions which are dual to Hölder functions. Stable homological and Markov eigenvalues yield analogous transverse structures to the unstable foliation of φ. The main tool for working with the homological spectrum is the Franks-Shub Theorem which holds for a general manifold and map. For the Markov spectrum we use the correspondence of the leaf space of stable foliation with a one-sided subshift of finite type. This identification allows the symbolic analog of a transverse arc function to be defined, analyzed, and applied.

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“…The semiconjugacy β 1 is a branched cover by [20] and so is locally a diffeomorphism at all but finitely many points and point inverses are finite sets. On the other hand, using the results above the semiconjugacy β 2 is Hölder exponent ν = log(µ)/ log(λ), but no larger ν.…”

Section: Examplementioning

confidence: 99%

Semiconjugacies to angle-doubling

Boyland

2005

Proc. Amer. Math. Soc.

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Abstract. A simple consequence of a theorem of Franks says that whenever a continuous map, g, is homotopic to angle-doubling on the circle, it is semiconjugate to it. We show that when this semiconjugacy has one disconnected point inverse, then the typical point in the circle has a point inverse with uncountably many connected components. Further, in this case the topological entropy of g is strictly larger than that of angle-doubling, and the semiconjugacy has unbounded variation. An analogous theorem holds for degree-D circle maps with D > 2.

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Pseudo-Anosov homeomorphisms with quadratic expansion

Cited by 13 publications

References 6 publications

Hyperbolic pseudo-Anosov maps almost everywhere embed into a toral automorphism

Hyperbolic pseudo-Anosov maps almost everywhere embed into a toral automorphism

On eigen-structures for pseudoAnosov maps

Semiconjugacies to angle-doubling

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