An infinite surface with the lattice property Ⅱ: Dynamics of pseudo-Anosovs

W. Patrick Hooper

doi:10.3934/jmd.2019009

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2019, Volume 14: 243-276. Doi: 10.3934/jmd.2019009

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An infinite surface with the lattice property Ⅱ: Dynamics of pseudo-Anosovs

W. Patrick Hooper^1,2,

1.
Department of Mathematics, The City College of New York, 160 Convent Ave, New York, NY 10031
2.
Department of Mathematics, The Graduate Center, CUNY, 365 5th Ave, New York, NY 10016, USA

Received: March 03, 2018

Revised: July 18, 2018

Published: March 2019

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Abstract

We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfies a mixing-type formula with polynomial decay. Then we consider the extent to which the action of these hyperbolic affine automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves and on homology classes. These objects are exponentially attracted by the expanding and contracting foliations but exhibit polynomial decay. We are able to work out exact asymptotics of these limiting quantities because of special integral formula for algebraic intersection number which is attuned to the geometry of the surface and its deformations.

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Mathematics Subject Classification: Primary: 37D40; Secondary: 37E30, 57M60.

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Figure 1. Veech's double 10-gon surface. Edge labels indicate glued edges

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Figure 2. From left to right, the surfaces $ {\mathbf{P}}_{\cos \frac{2 \pi}{9}} $, $ {\mathbf{P}}_1 $ and $ {\mathbf{P}}_{\frac{5}{4}} $ are shown

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Figure 3. The horizontal cylinders $ {\mathscr{A}}_i $ of $ {\mathbf{P}}_1 $

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Figure 6. Left: the surface $ {\mathbf{P}}_1 $ with saddle connections $ \sigma_i $ labeled. Right: the closed geodesic $ \gamma_{-3} $

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Figure 4. The fixed point sets of $ A_c $, $ B_c $ and $ C_c $ in the upper half plane model depicted from left to right for the cases of $ c = \cos\frac{\pi}{4} $, $ c = 1 $, and $ c = \frac{5}{4} $ from left to right

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Figure 5. A segment and its image under $ M_c $

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Figure 7. Two intersecting cylinders developed into the plane. Roman numerals indicate edge identifications, which reconstruct the cylinders

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References

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