Flat Surfaces. Marcelo Viana. CIM - Coimbra

Transcription

1 CIM - Coimbra

2 Main Reference Interval Exchange Transformations and Teichmüller Flows viana/out/ietf.pdf Rauzy, Keane, Masur, Veech, Hubbard, Kerckhoff, Smillie, Kontsevich, Zorich, Eskin, Nogueira, Rudolph, Forni, vila,...

3 Main Reference Interval Exchange Transformations and Teichmüller Flows viana/out/ietf.pdf Rauzy, Keane, Masur, Veech, Hubbard, Kerckhoff, Smillie, Kontsevich, Zorich, Eskin, Nogueira, Rudolph, Forni, vila,...

4 Outline 1 s 2 Interval exchanges Induction operator Teichmüller flow Genus 1 case 3 Invariant measures Unique ergodicity symptotic flag

5 Outline s 1 s 2 Interval exchanges Induction operator Teichmüller flow Genus 1 case 3 Invariant measures Unique ergodicity symptotic flag

6 s Consider any planar polygon with even number of sides, organized in pairs of parallel sides with the same length. Identify sides in the same pair, by translation. S E W N Two translation surfaces are the same if they are isometric.

7 s Structure of translation surfaces Riemann surface with a translation atlas holomorphic complex differential 1-form α z = dz flat Riemann metric with a unit parallel vector

8 s Structure of translation surfaces Riemann surface with a translation atlas holomorphic complex differential 1-form α z = dz flat Riemann metric with a unit parallel vector

9 s Structure of translation surfaces Riemann surface with a translation atlas holomorphic complex differential 1-form α z = dz flat Riemann metric with a unit parallel vector

10 Singularities s Points of the surface arising from the vertices of the polygon may correspond to (conical) singularities of the metric. PSfrag replacements V

11 Singularities s Points of the surface arising from the vertices of the polygon may correspond to (conical) singularities of the metric. V PSfrag replacements V

12 Singularities s Points of the surface arising from the vertices of the polygon may correspond to (conical) singularities of the metric. V V PSfrag replacements V

13 Singularities s Points of the surface arising from the vertices of the polygon may correspond to (conical) singularities of the metric. V V V PSfrag replacements V

14 Singularities s Points of the surface arising from the vertices of the polygon may correspond to (conical) singularities of the metric. V V V V PSfrag replacements V

15 Singularities s Points of the surface arising from the vertices of the polygon may correspond to (conical) singularities of the metric. V V V V V V PSfrag replacements V V

16 Singularities s In this example the neighborhood of V corresponds to gluing 8 copies of the angular sector of angle 3π/4. Thus, the total angle of the metric at V is 6π. This singularity corresponds to a zero of order 3 of the complex 1-form α.

17 Geodesics s We want to understand the behavior of geodesics with a given direction. In particular, When are the geodesics closed? When are they dense in the surface?

18 Motivation s quadratic differential on a Riemann surface assigns to each point a complex quadratic form from the tangent space, depending holomorphically on the point. In local coordinates z, it is given by some ϕ(z)dz 2 where ϕ(z) is a holomorphic function. The expression ψ(w)dw 2 relative to another local coordinate w satisfies ( ) dz 2 ψ(w) = ϕ(z) dw on the intersection of the domains.

19 Motivation s quadratic differential on a Riemann surface assigns to each point a complex quadratic form from the tangent space, depending holomorphically on the point. In local coordinates z, it is given by some ϕ(z)dz 2 where ϕ(z) is a holomorphic function. The expression ψ(w)dw 2 relative to another local coordinate w satisfies ( ) dz 2 ψ(w) = ϕ(z) dw on the intersection of the domains.

20 Motivation s Quadratic differentials play a central role in the theory of Riemann surfaces, in connection with understanding the deformations of the holomorphic structure. replacements V H vector v is vertical if q z (v) > 0 and it is horizontal if q z (v) < 0.

21 Motivation s The square of a holomorphic complex 1-form α z = φ(z)dz is always a quadratic differential. Every quadratic differential q is locally the square of a complex 1-form α. Moreover, we may find a two-to-one covering S S, ramified over some singularities, such that the lift of q to S is the square of a complex 1-form α, globally.

22 Motivation s The square of a holomorphic complex 1-form α z = φ(z)dz is always a quadratic differential. Every quadratic differential q is locally the square of a complex 1-form α. Moreover, we may find a two-to-one covering S S, ramified over some singularities, such that the lift of q to S is the square of a complex 1-form α, globally.

23 Calculating the genus PSfrag replacements B C s D D B C 2 2g = F + V

24 Calculating the genus PSfrag replacements B C s D D B C 2 2g = 4d + V, 2d = # sides

25 Calculating the genus PSfrag replacements B C s D D B C 2 2g = 4d 6d + V, 2d = # sides

26 Calculating the genus PSfrag replacements B C s D D B C 2 2g = 4d 6d + (d + κ + 1), κ = # singularities

27 Calculating the genus PSfrag replacements B C s D D B C 2g = d κ + 1, where 2d = # sides and κ = # singularities.

28 Computing the singularities s B C D E F G PSfrag replacements V H G F E D C B Let m + 1 = 1/2 the number of associated interior vertices: 2π(m + 1) = conical angle m = multiplicity of the singularity

29 Strata s Let m 1,..., m κ be the multiplicities of the singularities: (m 1 + 1) + + (m κ + 1) = d 1 = 2g + κ 2. Gauss-Bonnet formula m m κ = 2g 2 g (m 1,..., m κ ) denotes the space of all translation surfaces with κ singularities, having multiplicities m i.

30 Strata s Let m 1,..., m κ be the multiplicities of the singularities: (m 1 + 1) + + (m κ + 1) = d 1 = 2g + κ 2. Gauss-Bonnet formula m m κ = 2g 2 g (m 1,..., m κ ) denotes the space of all translation surfaces with κ singularities, having multiplicities m i.

31 Strata s Each stratum g (m 1,..., m κ ) is an orbifold of dimension 2d = 4g + 2κ 2. Local coordinates: PSfrag replacements ζ B ζ C ζ D ζ ζ D ζ B ζ C ζ ζ α = (λ α, τ α ) together with π = combinatorics of pairs of sides

32 Outline Interval exchanges Induction operator Teichmüller flow Genus 1 case 1 s 2 Interval exchanges Induction operator Teichmüller flow Genus 1 case 3 Invariant measures Unique ergodicity symptotic flag

33 Cross-sections to vertical flow PSfrag replacements Interval exchanges Induction operator Teichmüller flow Genus 1 case ζ B ζ C ζ D ζ ζ D ζ B ζ C ζ The return map of the vertical geodesic flow to some cross-section is an interval exchange transformation. To analyze the behavior of longer and longer geodesics, we consider return maps to shorter and shorter cross-sections.

34 Cross-sections to vertical flow PSfrag replacements Interval exchanges Induction operator Teichmüller flow Genus 1 case ζ B ζ C ζ D ζ ζ D ζ B ζ C ζ The return map of the vertical geodesic flow to some cross-section is an interval exchange transformation. To analyze the behavior of longer and longer geodesics, we consider return maps to shorter and shorter cross-sections.

35 Shortening the cross-section Interval exchanges Induction operator Teichmüller flow Genus 1 case B C D E ements E D C B

36 Shortening the cross-section Interval exchanges Induction operator Teichmüller flow Genus 1 case ements B C D E E D C B

37 Shortening the cross-section Interval exchanges Induction operator Teichmüller flow Genus 1 case ements B C D E B C D E E D C B E D C B

38 replacements Interval exchanges Interval exchanges Induction operator Teichmüller flow Genus 1 case B C D E E D C B Interval exchanges are described by combinatorial data ( ) B C D E π = E D C B and metric data λ = (λ, λ B, λ C, λ D, λ E ).

39 Rauzy induction ements Interval exchanges Induction operator Teichmüller flow Genus 1 case B C D π = ( B C D C B D C ) B D λ = (λ, λ B, λ C, λ D )

40 Rauzy induction Interval exchanges Induction operator Teichmüller flow Genus 1 case ements B C D π = ( B C D C B D C ) B D λ = (λ, λ B, λ C, λ D ) B C C B D This is a bottom case: of the two rightmost intervals, the bottom one is longest.

41 The Rauzy lgorithm Interval exchanges Induction operator Teichmüller flow Genus 1 case ements B C D π = π = ( B C D C B D ( D B C C B D C ) C ) D B D λ = (λ, λ B, λ C, λ D ) B C B D λ = (λ λ D, λ B, λ C, λ D )

42 Teichm üller flow Interval exchanges Induction operator Teichmüller flow Genus 1 case The Teichmüller flow is the action induced on the stratum by the diagonal subgroup of SL(2, R). In coordinates: (π, λ, τ) (π, e t λ, e t τ) t Both the Teichmüller flow and the induction operator preserve the area. We always suppose area 1.

43 Teichm üller flow Interval exchanges Induction operator Teichmüller flow Genus 1 case The Teichmüller flow is the action induced on the stratum by the diagonal subgroup of SL(2, R). In coordinates: (π, λ, τ) (π, e t λ, e t τ) t Both the Teichmüller flow and the induction operator preserve the area. We always suppose area 1.

44 Rauzy renormalization Interval exchanges Induction operator Teichmüller flow Genus 1 case The Rauzy renormalization operator is the composition of the Rauzy induction (this reduces the width of the surface/length of the cross-section) the Teichmüller (the right time to restore the width of the surface/length of the cross-section back to the initial value) It is defined both on the space of translation surfaces R : (π, λ, τ) (π, λ, τ ) and on the space of interval exchange transformations R : (π, λ) (π, λ ).

45 Flat torus Interval exchanges Induction operator Teichmüller flow Genus 1 case B PSfrag replacements B d = 2 κ = 1 m = 0 (removable!) g = 1

46 Exchanges of two intervals Interval exchanges Induction operator Teichmüller flow Genus 1 case replacements B B

47 Exchanges of two intervals Interval exchanges Induction operator Teichmüller flow Genus 1 case replacements 1 x x x 1 x

48 Exchanges of two intervals Interval exchanges Induction operator Teichmüller flow Genus 1 case 1 2x x replacements x 1 2x

49 Exchanges of two intervals Interval exchanges Induction operator Teichmüller flow Genus 1 case (1 2x)/(1 x) replacements x/(1 x) x/(1 x) (1 2x)/(1 x)

50 Renormalization for d = 2 Interval exchanges Induction operator Teichmüller flow Genus 1 case R(x) = { x/(1 x) for x (0, 1/2) 2 1/x for x (1/2, 1). PSfrag replacements 0 1/2 1

51 Outline Invariant measures Unique ergodicity symptotic flag 1 s 2 Interval exchanges Induction operator Teichmüller flow Genus 1 case 3 Invariant measures Unique ergodicity symptotic flag

52 Invariant measures Invariant measures Unique ergodicity symptotic flag Theorem (Masur, Veech, 1982) There is a natural finite volume measure on each stratum, invariant under the Teichmüller flow. This measure is ergodic (restricted to the hypersurface of surfaces with unit area). Ergodic means that almost every orbit spends in each subset of the stratum a fraction of the time equal to the volume of the subset. In particular, almost every orbit is dense in the stratum.

53 Invariant measures Invariant measures Unique ergodicity symptotic flag In coordinates, this invariant volume measure is given by dπdλdτ where dπ is the counting measure Theorem (Veech, 1982) The renormalization operator R admits an absolutely continuous invariant measure absolutely continuous with respect to dλ. This measure is ergodic and unique up to rescaling.

54 Invariant measures Invariant measures Unique ergodicity symptotic flag In coordinates, this invariant volume measure is given by dπdλdτ where dπ is the counting measure Theorem (Veech, 1982) The renormalization operator R admits an absolutely continuous invariant measure absolutely continuous with respect to dλ. This measure is ergodic and unique up to rescaling.

55 n example Invariant measures Unique ergodicity symptotic flag For d = 2, the operator R is given by PSfrag replacements 0 1/2 1 The absolutely continuous invariant measure is infinite!

56 Minimality Invariant measures Unique ergodicity symptotic flag Theorem (Keane, 1975) If λ is rationally independent then the interval exchange defined by (π, λ) is minimal: every orbit is dense.

57 Keane conjecture Invariant measures Unique ergodicity symptotic flag Theorem (Masur, Veech, 1982) For every π and almost every λ, the interval exchange defined by (π, λ) is uniquely ergodic: it admits a unique invariant probability. This result is a consequence of the ergodicity of the renormalization operator.

58 Unique ergodicity Invariant measures Unique ergodicity symptotic flag Theorem (Kerckhoff, Masur, Smillie, 1986) For every polygon, and almost every direction, the (translation) geodesic flow in that direction is uniquely ergodic.

59 Linear cocycles Invariant measures Unique ergodicity symptotic flag

60 References Invariant measures Unique ergodicity symptotic flag Keane, M., Interval exchange transformations, Math. Zeit. 141 (1975), Kontsevich, M.; Zorich,. Connected components of the moduli spaces of belian differentials with prescribed singularities. Invent. Math. 153 (2003), Ledrappier, F. Positivity of the exponent for stationary sequences of matrices. Lyapunov exponents, Lecture Notes in Math (1986), Masur, H. Interval exchange transformations and measured foliations. nn. of Math. 115 (1982), Oseledets, V. I. multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), Rauzy, G. Echanges d intervalles et transformations induites. cta rith. 34 (1979), Schwartzman, S. symptotic cycles. nn. of Math. 66 (1957), Veech, W. Gauss measures for transformations on the space of interval exchange maps. nn. of Math. 115 (1982), Zorich,. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. nn. Inst. Fourier 46 (1996), Zorich,. How do the leaves of a closed 1-form wind around a surface? Pseudoperiodic topology, mer. Math. Soc. Transl. Ser. 2, 197 (1999),