February  2019, 14: 243-276. doi: 10.3934/jmd.2019009

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

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 04, 2018 Revised  July 19, 2018 Published  March 2019

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.

Citation: W. Patrick Hooper. An infinite surface with the lattice property Ⅱ: Dynamics of pseudo-Anosovs. Journal of Modern Dynamics, 2019, 14: 243-276. doi: 10.3934/jmd.2019009
References:
[1]

J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997, 59–115.

[2]

V. Delecroix and W. P. Hooper, Flat Surfaces in Sage, https://github.com/videlec/sage-flatsurf, accessed Feb. 22, 2018.

[3] A. FathiF. Laudenbach and V. Poénaru, Thurston's Work on Surfaces, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012.
[4]

W. Patrick Hooper, An infinite surface with the lattice property. Ⅰ: Veech groups and coding geodesics, Trans. Am. Math. Soc., 366 (2014), 2625-2649. doi: 10.1090/S0002-9947-2013-06139-9.

[5]

W. Patrick Hooper, An infinite surface with the lattice property. I: Veech groups and coding geodesics, Trans. Am. Math. Soc., 366 (2014), 2625-2649. doi: 10.1090/S0002-9947-2013-06139-9.

[6]

W. Patrick Hooper, The invariant measures of some infinite interval exchange maps, Geometry & Topology, 19 (2015), 1895-2038. doi: 10.2140/gt.2015.19.1895.

[7] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York-London, 1974.
[8]

W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), no. 2, 417–431. doi: 10.1090/S0273-0979-1988-15685-6.

[9]

W. P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, in Low-dimensional Topology and Kleinian Groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., 112, Cambridge Univ. Press, Cambridge, 1986, 91–112.

[10]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

show all references

References:
[1]

J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997, 59–115.

[2]

V. Delecroix and W. P. Hooper, Flat Surfaces in Sage, https://github.com/videlec/sage-flatsurf, accessed Feb. 22, 2018.

[3] A. FathiF. Laudenbach and V. Poénaru, Thurston's Work on Surfaces, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012.
[4]

W. Patrick Hooper, An infinite surface with the lattice property. Ⅰ: Veech groups and coding geodesics, Trans. Am. Math. Soc., 366 (2014), 2625-2649. doi: 10.1090/S0002-9947-2013-06139-9.

[5]

W. Patrick Hooper, An infinite surface with the lattice property. I: Veech groups and coding geodesics, Trans. Am. Math. Soc., 366 (2014), 2625-2649. doi: 10.1090/S0002-9947-2013-06139-9.

[6]

W. Patrick Hooper, The invariant measures of some infinite interval exchange maps, Geometry & Topology, 19 (2015), 1895-2038. doi: 10.2140/gt.2015.19.1895.

[7] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York-London, 1974.
[8]

W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), no. 2, 417–431. doi: 10.1090/S0273-0979-1988-15685-6.

[9]

W. P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, in Low-dimensional Topology and Kleinian Groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., 112, Cambridge Univ. Press, Cambridge, 1986, 91–112.

[10]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

Figure 1.  Veech's double 10-gon surface. Edge labels indicate glued edges
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
Figure 3.  The horizontal cylinders $ {\mathscr{A}}_i $ of $ {\mathbf{P}}_1 $
Figure 6.  Left: the surface $ {\mathbf{P}}_1 $ with saddle connections $ \sigma_i $ labeled. Right: the closed geodesic $ \gamma_{-3} $
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
Figure 5.  A segment and its image under $ M_c $
Figure 7.  Two intersecting cylinders developed into the plane. Roman numerals indicate edge identifications, which reconstruct the cylinders
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