Expanding factors for pseudo-Anosov homeomorphisms.

Rykken, Elyn

doi:10.1307/mmj/1030132411

Cited by 15 publications

(16 citation statements)

References 4 publications

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“…Since f * 1 : H 1 (M) → H 1 (M) and a Markov matrix will have the same eigenvalues up to zeros and roots of unity [8], λ is the unique eigenvalue with largest modulus. An element of H 1 (M ) can be expressed as a linear combination of these basis elements.…”

Section: Lemma 22 There Exists a δ > 0 Such That Given Any Rectanglmentioning

confidence: 99%

Pseudo-Anosov homeomorphisms with quadratic expansion

Franks

Rykken

1999

Proc. Amer. Math. Soc.

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Abstract. We show that if f : M → M is a pseudo-Anosov homeomorphism on an orientable surface with oriented unstable manifolds and a quadratic expanding factor, then there is a hyperbolic toral automorphism on T 2 and a map h : M → T 2 such that h is a semi-conjugacy and (M, h) is a branched covering space of T 2 . We also give another characterization of pseudo-Anosov homeomorphisms with quadratic expansion in terms of the kinds of Euclidean foliations they admit which are compatible with the affine structure associated to f .

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Section: Lemma 22 There Exists a δ > 0 Such That Given Any Rectanglmentioning

confidence: 99%

Pseudo-Anosov homeomorphisms with quadratic expansion

Franks

Rykken

1999

Proc. Amer. Math. Soc.

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“…The author noticed that a similar proof was given in [4]. Theorem 1.6 (Rykken [19]). Our following results are for the case when (F u , µ u ) has singularities other than unpunctured 3-pronged and punctured 1-pronged singularities.…”

Section: Resultsmentioning

confidence: 61%

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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“…A proof of the fact that any affine automorphism lifts can be found in [Ry,Theorem 3.2]. The statement about the number of cylinders and rectangles follows immediately from the fact that the degree of the cover is 2.…”

Section: Taking Orientation Double Covers Ismentioning

confidence: 99%

Finiteness results for flat surfaces: large cusps and short geodesics

Smillie¹,

Weiss²

2010

Comment. Math. Helv.

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Abstract. For fixed g and T we show the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a cusp of hyperbolic co-area less than T . We obtain new restrictions on Veech groups: any non-elementary Fuchsian group can appear only finitely many times in a fixed stratum; any non-elementary Veech group is of finite index in its normalizer; and the quotient of H by a non-lattice Veech group admits arbitrarily large embedded disks. A key ingredient of the proof is the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a hyperbolic element with eigenvalue less than T . Mathematics Subject Classification (2000). 57M50, 37D40.

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Expanding factors for pseudo-Anosov homeomorphisms.

Cited by 15 publications

References 4 publications

Pseudo-Anosov homeomorphisms with quadratic expansion

Pseudo-Anosov homeomorphisms with quadratic expansion

Dynnikov and train track transition matrices of pseudo-Anosov braids

Finiteness results for flat surfaces: large cusps and short geodesics

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