Dynamics on some Z 2 -covers of half-translation surfaces

Transcription

1 Dynamics on some Z 2 -covers of half-translation surfaces Chris Johnson Clemson University April 27, 2011

2 Outline Background The windtree model HLT s recurrence result The folded plane Panov s density result Generalizing Our results

3 The flat torus

4 The flat torus A B

5 The flat torus A B

6 The flat torus A B

7 The flat torus B B B B A B B B B B

8 The flat torus B B B B A B B B B B

9 The flat torus

10 The flat torus Theorem (Weil) On the flat torus a line will be Closed iff the slope is rational Dense iff the slope is irrational

11 Translation surfaces Euclidean polygons glued together by translation along parallel edges of the same length A surface together with an atlas where, away from a discrete set of points, coordinate changes are accomplished by translation; z z + c A Riemann surface together with a holomorphic 1-form.

12 Translation surfaces Euclidean polygons glued together by translation along parallel edges of the same length A surface together with an atlas where, away from a discrete set of points, coordinate changes are accomplished by translation; z z + c A Riemann surface together with a holomorphic 1-form.

13 Translation surfaces Euclidean polygons glued together by translation along parallel edges of the same length A surface together with an atlas where, away from a discrete set of points, coordinate changes are accomplished by translation; z z + c A Riemann surface together with a holomorphic 1-form.

14 Half-translation surfaces Euclidean polygons glued together by translation posibly preceeded by a 180 -rotation along parallel edges of the same length A surface together with an atlas where, away from a discrete set of points, coordinate changes are accomplished by z ±z + c A Riemann surface together with a quadratic differential.

15 Half-translation surfaces Euclidean polygons glued together by translation posibly preceeded by a 180 -rotation along parallel edges of the same length A surface together with an atlas where, away from a discrete set of points, coordinate changes are accomplished by z ±z + c A Riemann surface together with a quadratic differential.

16 Half-translation surfaces Euclidean polygons glued together by translation posibly preceeded by a 180 -rotation along parallel edges of the same length A surface together with an atlas where, away from a discrete set of points, coordinate changes are accomplished by z ±z + c A Riemann surface together with a quadratic differential.

17 The action of SL 2 (R) The group SL 2 (R) acts on the set of translation surfaces by deforming the polygons. The stabilizer of a surface S under this action is called the Veech group of S and denoted Γ(S).

18 The action of SL 2 (R) [ ]

19 The action of SL 2 (R) [ ]

20 The action of SL 2 (R) [ ]

21 The action of SL 2 (R) [ ]

22 Theorem (Veech, 1989) Let S is a translation surface and Γ(S) its Veech group. If Γ(S) is a lattice subgroup of SL 2 (R) (i.e., SL 2 (R)Γ(S) has finite volume wrt the Haar measure), then for any direction, all regular trajectories in that direction are periodic, or all regular trajectories in that direction are dense. This property is called the Veech dichotomy, and surfaces satisfying the Veech dichotomy are called Veech surfaces.

23 Theorem (Veech, 1989) Let S is a translation surface and Γ(S) its Veech group. If Γ(S) is a lattice subgroup of SL 2 (R) (i.e., SL 2 (R)Γ(S) has finite volume wrt the Haar measure), then for any direction, all regular trajectories in that direction are periodic, or all regular trajectories in that direction are dense. This property is called the Veech dichotomy, and surfaces satisfying the Veech dichotomy are called Veech surfaces.

24 The natural maps on a translation surface are the affine diffeomorphisms, which are maps which locally have the form z Az + b where A SL 2 (R). Because coordinate changes are accomplished by translation, the A above is independent of the choice of coordinates. We can define a derivative map D : Aff(S) SL 2 (R). Theorem (Veech, 1989) Γ(S) = D(Aff(S)).

25 The natural maps on a translation surface are the affine diffeomorphisms, which are maps which locally have the form z Az + b where A SL 2 (R). Because coordinate changes are accomplished by translation, the A above is independent of the choice of coordinates. We can define a derivative map D : Aff(S) SL 2 (R). Theorem (Veech, 1989) Γ(S) = D(Aff(S)).

26 The natural maps on a translation surface are the affine diffeomorphisms, which are maps which locally have the form z Az + b where A SL 2 (R). Because coordinate changes are accomplished by translation, the A above is independent of the choice of coordinates. We can define a derivative map D : Aff(S) SL 2 (R). Theorem (Veech, 1989) Γ(S) = D(Aff(S)).

27 Dehn twists

28 Cylinder decompositions We can decompose the surface into two cylinders. Can perform Dehn twists on these cylinders. Performing the same twist on each cylinder gives a globally defined affine map. (Can only perform the same twists if the ratio of width to height of one cylinder is a rational multiple of the other.)

29 Cylinder decompositions We can decompose the surface into two cylinders. Can perform Dehn twists on these cylinders. Performing the same twist on each cylinder gives a globally defined affine map. (Can only perform the same twists if the ratio of width to height of one cylinder is a rational multiple of the other.)

30 Cylinder decompositions We can decompose the surface into two cylinders. Can perform Dehn twists on these cylinders. Performing the same twist on each cylinder gives a globally defined affine map. (Can only perform the same twists if the ratio of width to height of one cylinder is a rational multiple of the other.)

31 The windtree model

32 The windtree model

33 The windtree model

34 The windtree model The plane with rectangles removed periodically defines the billiard table T α,β denotes the table obtained by removing α β rectangles The ray represents the motion of a billiard ball Initially considered by Ehrenfest

35 The windtree model Unfolding the billiard Unfolding is a standard tool in billiard theory Take one copy of the table for each of the four possible reflections Glue the copies together at the edges This gives a surface which is a cover of the table The straight line flow on the surface projects to the billiard flow on the table

36 The windtree model Unfolding the billiard NW NE SW SE

37 The windtree model Unfolding the billiard NW NE SW SE

38 The windtree model Unfolding the billiard NW NE SW SE

39 The windtree model Unfolding the billiard NW NE SW SE

40 NW The windtree model Unfolding the billiard NE SW SE

41 The windtree model Unfolding the billiard We ll denote this unfolded surface U α,β This is a four-sheeted cover of T α,β... and a Z 2 -cover of the following genus 5 surface

42 The windtree model Unfolding the billiard Four L s surface is a 4-sheeted cover of These are genus 2 surfaces and are fairly well-understood. McMullen showed these surfaces are Veech (have optimal dynamics) iff the width and height are particular algebraic numbers: α = x + z d β = 1 x + z d

43 The windtree model HLT s recurrence result Theorem (Hubert, Lelievre, & Troubetzkoy, 2009) For α, β Q, where α = p q, β = r s with (p, q) = (r, s) = 1 and p, r odd, q, s even, every trajectory in almost every direction of U α,β is recurrent. Theorem (Hubert & Avila, 2011) For any (α, β) (0, 1) 2, almost every direction of U α,β is recurrent.

44 The windtree model HLT s recurrence result Theorem (Hubert, Lelievre, & Troubetzkoy, 2009) For α, β Q, where α = p q, β = r s with (p, q) = (r, s) = 1 and p, r odd, q, s even, every trajectory in almost every direction of U α,β is recurrent. Theorem (Hubert & Avila, 2011) For any (α, β) (0, 1) 2, almost every direction of U α,β is recurrent.

45 The folded plane

46 The folded plane Place horizontal line segments periodically

47 The folded plane Slit along line segments M N L R M S

48 The folded plane Identify Eastern and Western halves M N L R M S

49 The folded plane Each pair of adjacent slits produces a pillowcase Topologically this is a sphere with a hole in it So this construction gives the connected sum of the plane with a bunch of spheres (Which is just the plane)

50 The folded plane Behavior of lines

51 The folded plane Behavior of lines

52 The folded plane Behavior of lines d d

53 The folded plane Behavior of lines

54 The folded plane Sample trajectory

55 The folded plane Sample trajectory

56 The folded plane Sample trajectory

57 The folded plane Sample trajectory

58 The folded plane Sample trajectory

59 The folded plane Half-translation torus The folded plane (say C α for slits of length α) is a Z 2 cover of the following surface. This is a torus with a pillowcase attached Because of the cone points of angle π, this is a half-translation torus

60 The folded plane Panov s density result Theorem (Panov) For countably many directions, the line foliation of C α in that direction which has a dense leaf.

61 Generalizing Panov s construction The twice-folded torus An obvious generalization of Panov is to attach two pillowcases to the torus. This is again just a half-translation torus.

62 Generalizing Panov s construction The twice-folded plane Since we have a torus, its universal cover is the plane. However the pillowcases change the geometry of the plane.

63 Generalizing Panov s construction The twice-folded plane Since we have a torus, its universal cover is the plane. However the pillowcases change the geometry of the plane.

64 Generalizing Panov s construction The twice-folded plane Since we have a torus, its universal cover is the plane. However the pillowcases change the geometry of the plane.

65 Generalizing Panov s construction The twice-folded plane Since we have a torus, its universal cover is the plane. However the pillowcases change the geometry of the plane.

66 Generalizing Panov s result We d like to apply Panov s density argument to this twice-folded plane, but need a few things. Need for our twice-folded torus to be Veech Need a pseudo-anosov map whose action on homology is a 90 -rotation.

67 Generalizing Panov s result We d like to apply Panov s density argument to this twice-folded plane, but need a few things. Need for our twice-folded torus to be Veech Need a pseudo-anosov map whose action on homology is a 90 -rotation.

68 The orientation covering > > = > <

69 The orientation covering > > = > <

70 The orientation covering > > =

71 The orientation covering > > =

72 The orientation covering F ± α,β (g = 3) L α,β (g = 2) F α,β (g = 1)

73 The orientation covering Theorem (Vasilyev, 2005) Let p : R S be a ramified finite covering of Riemann surfaces branched over a finite set B. If τ is a quadratic differential on S, then SL(S, τ, B) and SL(R, p (τ)) are commensurate. Corollary (J. & Schmoll) F α,β is a Veech surface if and only if L α,β is Veech.

74 The orientation covering Theorem (Vasilyev, 2005) Let p : R S be a ramified finite covering of Riemann surfaces branched over a finite set B. If τ is a quadratic differential on S, then SL(S, τ, B) and SL(R, p (τ)) are commensurate. Corollary (J. & Schmoll) F α,β is a Veech surface if and only if L α,β is Veech.

75 A first result Theorem (J. & Schmoll) If α and β are of the form α = β = d 1 + k 2 d 1 k 2 where k, d N 0 and d 1 (mod 8), then the trajectory starting at a preimage of a pillowcase singularity in the eigendirection of the pseudo-anosov map is dense on C α,β.

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79 What does the same direction give on the corresponding windtree model?

80 What does the same direction give on the corresponding windtree model?

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86 Theorem (Delecroix, 2011) Any direction whose slope has can be written as a continued fraction consisting of all even numbers (e.g., 2 1 = [2, 2, 2,...]), the trajectory in the windtree model escapes. (Our direction above is 7 = [2; 1, 1, 1, 4].)

87 Theorem (Delecroix, 2011) Any direction whose slope has can be written as a continued fraction consisting of all even numbers (e.g., 2 1 = [2, 2, 2,...]), the trajectory in the windtree model escapes. (Our direction above is 7 = [2; 1, 1, 1, 4].)

88 The coverings U α,β C ± α,β 4 Z 2 2 T α,β 4L α,β 2 F ± α,β C α,β Z 2 L α,β F α,β

89 Where to go from here? Is the escaping trajectory a fractal? What is it that makes dense trajectories in C α,β escape in U α,β? Is there a dense direction in C α,β which is also dense on U α,β? Compare the ideas of Panov with the ergodicity results of Hooper. Want to get a general idea about recurrence and ergodicity of Z 2 -covers of compact translation surfaces.