Lyapunov exponents of Teichmüller flows

Lyapunov exponents ofteichmüller flows p 1/6 Lyapunov exponents of Teichmüller flows Marcelo Viana IMPA - Rio de Janeiro

Lyapunov exponents ofteichmüller flows p 2/6 Lecture # 1 Geodesic flows on translation surfaces

Abelian differentials Lyapunov exponents ofteichmüller flows p 3/6 Abelian differential = holomorphic -form (compact) Riemann surface on a

Abelian differentials Lyapunov exponents ofteichmüller flows p 3/6 Abelian differential = holomorphic -form (compact) Riemann surface on a Adapted local coordinates: then near a zero with multiplicity : then Adapted coordinates form a translation atlas: coordinate changes near any regular point have the form

Translation surfaces Lyapunov exponents ofteichmüller flows p 4/6 The translation atlas defines a flat metric with a finite number of conical singularities: frag replacements a parallel unit vector field (the upward direction) on the complement of the singularities Conversely, the flat metric and the parallel unit vector field characterize the translation structure completely

Geometric representation Lyapunov exponents ofteichmüller flows p 5/6 Consider any planar polygon with even number of sides, organized in pairs of parallel sides with the same length S E W N PSfrag replacements Identifying sides in the same pair, by translation, yields a translation surface Singularities arise from the vertices Every translation surface can be represented in this way, but not uniquely

Moduli spaces Lyapunov exponents ofteichmüller flows p 6/6 moduli space of Abelian differentials on Riemann surfaces of genus a complex orbifold of dimension a fiber bundle over the moduli space surfaces of genus of Riemann

Teichmüller geodesic flow Lyapunov exponents ofteichmüller flows p 7/6 The Teichmüller flow is the natural action bundle by the diagonal subgroup of on the fiber : Geometrically: eplacements The area of the surface is constant on orbits of this flow

Ergodicity Lyapunov exponents ofteichmüller flows p 8/6 Masur, Veech: Each stratum of carries a canonical volume measure, which is finite and invariant under the Teichmüller flow The Teichmüller flow is ergodic on every connected component of every stratum, restricted to any hypersurface of constant area Kontsevich, Zorich: Classified the connected components of all strata Each stratum has at most connected components Eskin, Okounkov, Pandharipande Computed the volumes of all the strata

Lyapunov exponents Lyapunov exponents ofteichmüller flows p 9/6 Fix a connected component of a stratum The Lyapunov spectrum of the Teichmüller flow has the form Theorem (Avila, Viana) Conjectured by Zorich, Kontsevich Veech proved Forni proved (flow is non-uniformly hyperbolic) (exactly exponents Before discussing the proof, let us describe an application )

Asymptotic cycles Lyapunov exponents ofteichmüller flows p 10/6 Sfrag Given replacements any long geodesic segment close it to get an element of in a given direction, : Kerckhoff, Masur, Smillie: The geodesic flow in almost every direction is uniquely ergodic Then converges uniformly to some when the length, where the asymptotic cycle depends only on the surface and the direction

# Asymptotic flag in homology of Corollary (Zorich) There are subspaces, such that with for all has amplitude from the deviation of "! ) genus of is bounded ( from the deviation of PSfrag replacements Lyapunov exponents ofteichmüller flows p 11/6

Zippered rectangles Lyapunov exponents ofteichmüller flows p 12/6 Almost every translation surface may be represented in the form of zippered rectangles (minimal number of rectangles required is ): PSfrag replacements

Coordinates in the stratum Lyapunov exponents ofteichmüller flows p 13/6 This defines local coordinates in the stratum: describes the combinatorics of the associated interval exchange map are the horizontal coordinates of the saddle-connections (= widths of the rectangles) are the vertical components of the saddle-connections has complex dimension The heights linear functions of of the rectangles are

Poincaré return map - 1st step Lyapunov exponents ofteichmüller flows p 14/6 We consider the return map of the Teichmüller flow to the cross section PSfrag replacements

Poincaré return map - 1st step Lyapunov exponents ofteichmüller flows p 15/6 We consider the return map of the Teichmüller flow to the cross section PSfrag replacements In this example: the top rectangle is the widest of the two

Poincaré return map - 1st step Lyapunov exponents ofteichmüller flows p 16/6 We consider the return map of the Teichmüller flow to the cross section PSfrag replacements

Rauzy-Veech induction Lyapunov exponents ofteichmüller flows p 17/6 This 1st step corresponds to, with (top case) except that same for except that Write where defined in this way Then is the linear operator and analogously for

Poincaré return map - 2nd step Lyapunov exponents ofteichmüller flows p 18/6 We consider the return map of the Teichmüller flow to the cross section PSfrag replacements

Poincaré return map - 2nd step Lyapunov exponents ofteichmüller flows p 19/6 We consider the return map of the Teichmüller flow to the cross section PSfrag replacements

Rauzy-Veech renormalization Lyapunov exponents ofteichmüller flows p 20/6 This 2nd step corresponds to with, and where is the normalizing factor The Poincaré return map (invertible) Rauzy-Veech renormalization is called

Rauzy-Veech cocycle and long geodesics Lyapunov exponents ofteichmüller flows p 21/6 Consider the linear cocycle over the map defined by

Rauzy-Veech cocycle and long geodesics Lyapunov exponents ofteichmüller flows p 21/6 Consider the linear cocycle over the map defined by PSfrag replacements Consider a geodesic segment crossing each rectangle vertically Close it by joining the endpoints to some fixed point in the cross-section This defines some

Rauzy-Veech cocycle and long geodesics Lyapunov exponents ofteichmüller flows p 22/6 Consider the linear cocycle over the map defined by PSfrag replacements

Rauzy-Veech cocycle and long geodesics Lyapunov exponents ofteichmüller flows p 22/6 Consider the linear cocycle over the map defined by PSfrag replacements except that In other words, builds long geodesics

Lyapunov exponents and asymptotic flag Lyapunov exponents ofteichmüller flows p 23/6 So, it seems likely that the behavior of long geodesics be described by the asymptotics of the Rauzy-Veech cocycle, the flag being defined by the Oseledets subspaces of with positive Lyapunov exponents the Lyapunov exponents of (for an appropriate invariant measure) be related to the Lyapunov exponents of the Teichmüller flow This suggests we should try to prove that has positive Lyapunov exponents and they are all distinct There is one technical difficulty, however:

Zorich cocycle Lyapunov exponents ofteichmüller flows p 24/6 The Rauzy-Veech renormalization does have a natural invariant measure (on each stratum) related to the invariant volume of the Teichmüller flow But this measure is infinite Zorich introduced an accelerated renormalization and an accelerated cocycle and where is smallest such that the Rauzy iteration changes from top to bottom or vice-versa This map has a natural invariant volume probability (on each stratum) and this probability is ergodic

Zorich cocycle Lyapunov exponents ofteichmüller flows p 25/6 The Lyapunov spectrum of the Zorich cocycle form has the and it is related to the spectrum of the Teichmüller flow by

Zorich cocycle Lyapunov exponents ofteichmüller flows p 25/6 The Lyapunov spectrum of the Zorich cocycle form has the and it is related to the spectrum of the Teichmüller flow by Theorem (Avila, Viana) Forni: proved (cocycle is non-uniformly hyperbolic)

Lyapunov exponents ofteichmüller flows p 26/6 Lecture # 2 Simplicity criterium for Lyapunov spectra

Linear cocycles Lyapunov exponents ofteichmüller flows p 27/6 A linear cocycle over a map is an extension where with Note Let be an -invariant ergodic probability on such that Then the Oseledets theorem applies We say has simple Lyapunov spectrum if all subspaces in the Oseledets decomposition are -dimensional

Hypotheses Assume that is the shift map on, where the alphabet is at most countable Denote for all for all Denote by the cylinder of sequences such that for all Lyapunov exponents ofteichmüller flows p 28/6

Hypotheses Assume that the ergodic probability has bounded distortion: is positive on cylinders and there exists such that for every and every Lyapunov exponents ofteichmüller flows p 29/6

Hypotheses Assume that the ergodic probability has bounded distortion: is positive on cylinders and there exists such that for every and every is locally constant: More generally: is continuous and admits stable and unstable holonomies Lyapunov exponents ofteichmüller flows p 29/6

Pinching Lyapunov exponents ofteichmüller flows p 30/6 Let be the monoid generated by the We call the cocycle (pinching) if for every such that the -eccentricity there exists some, for every ents 1

Twisting, be the monoid generated by the We call the cocycle Let and any finite family there exists some for all twisting if for any such that Lyapunov exponents ofteichmüller flows p 31/6

Twisting Lyapunov exponents ofteichmüller flows p 31/6 Let be the monoid generated by the We call the cocycle, twisting if for any such that and any finite family there exists some for all simple if is both pinching and twisting Lemma is simple if and only if is simple

Criterium for simplicity Lyapunov exponents ofteichmüller flows p 32/6 Theorem 1 (Bonatti-V, Avila-V) If the cocycle Lyapunov spectrum is simple is simple then its Guivarc h, Raugi obtained a simplicity condition for products of independent random matrices, and this was improved by Gol dsheid, Margulis Later we shall apply this criterium to the Zorich cocycles Right now let us describe some main ideas in the proof

Comments on the strategy Lyapunov exponents ofteichmüller flows p 33/6 If the first exponents are strictly larger, the corresponding subbundle is an attractor for the action of on the Grassmannian bundle : ag replacements Notice that is constant on local unstable sets

Invariant section Lyapunov exponents ofteichmüller flows p 34/6 Proposition 1 Fix exists a measurable section Assume the cocycle is simple Then there such that 1 is constant on local unstable sets and -invariant, that is, at -almost every point 2 expanded -subspace converges to and the image of the most 3 for any at -almost every point g replacements, we have that is transverse to

Proof of Theorem 1 Lyapunov exponents ofteichmüller flows p 35/6 Applying the proposition to the inverse cocycle and to the dimension, we find another invariant section with similar properties for Recall is simple if and only if is simple Keeping in mind that is constant on unstable sets and is constant on stable sets, part 3 implies that at -almost every point

Lyapunov exponents ofteichmüller flows p 36/6 Proof of PSfrag replacements Suppose the claim fails on a set with By local product structure, there exist points has positive -measure, where Define Then is not transverse to on such that for all and This contradicts part 3

Proof of Theorem 1 Lyapunov exponents ofteichmüller flows p 37/6 lacements From part 2 we get that the Lyapunov exponents of are strictly larger than the ones of Using that is close to for large, one finds an invariant cone field around for the cocycle induced by on some convenient subset (with large return times) Hence, there are Lyapunov exponents (for either cocycle) which are larger than the other ones Theorem 1 follows

( 4555 / -states -state if it projects is a such that A probability on down to and there is - *,+ ) and &"' & # "% $# "! for every 7 ;$< 4555 1; 8 4 7 2$: 4555 1298 4 replacements /0 7 6 2 0 12$34 Lyapunov exponents ofteichmüller flows p 38/6

-states Lyapunov exponents ofteichmüller flows p 39/6 Equivalently, is a -state iff it may be disintegrated as on - * + where is equivalent to whenever derivative uniformly bounded by, with

-states Lyapunov exponents ofteichmüller flows p 39/6 Equivalently, is a -state iff it may be disintegrated as on - * + where is equivalent to whenever derivative uniformly bounded by, with -states always exist: for instance, probability in the Grassmannian any Cesaro limit of iterates of a under the cocycle - * + for any -state is invariant - * +

Proof of Proposition 1 Lyapunov exponents ofteichmüller flows p 40/6 Proposition 2 Let - * + Then the support of Grassmannian the push-forwards be an invariant -state and be its projection to is not contained in any hyperplane section of the of under surely to a Dirac measure at some point frag replacements converge almost 0

- Proof of Proposition 1 Lyapunov exponents ofteichmüller flows p 41/6 Proposition 1 follows from Proposition 2 together with Lemma Let be a probability measure on hyperplane section be a sequence of linear isomorphisms and - *,+ which is not supported in a - * + - * + If the push-forwards converge to a Dirac measure eccentricity and the images expanded -subspace converge to - then the of the most

Proof of Proposition 2 Lyapunov exponents ofteichmüller flows p 42/6 Part 1: is not supported in any hyperplane section This follows from the twisting hypothesis Part 2: the sequence converges to a Dirac measure The proof has a few steps The most delicate is to show that some subsequence converges to a Dirac measure:

Proof of Proposition 2 Lyapunov exponents ofteichmüller flows p 43/6 For almost every there exist converges to a Dirac measure such that

Proof of Proposition 2 /0 / # Lyapunov exponents ofteichmüller flows p 43/6 For almost every there exist converges to a Dirac measure such that By hypothesis, there is some strongly pinching The are chosen so that for some convenient and large This uses ergodicity % # ' # g replacements 0

Proof of Proposition 2 Lyapunov exponents ofteichmüller flows p 44/6 Let denote the projection to the Grassmannian of the normalized restriction of to the cylinder that contains The sequence probability on converges almost surely to some - * + This follows from a simple martingale argument

Lyapunov exponents ofteichmüller flows p 45/6 Lecture # 3 Zorich cocycles are pinching and twisting

Strata and Rauzy classes Lyapunov exponents ofteichmüller flows p 46/6 On each stratum local coordinates, where we introduced a system of # is a pair of permutations of the alphabet and live in convenient subsets of, The pair determines the stratum and even its connected component An (extended) Rauzy class is the set of all corresponding to a given connected component of a stratum (there is also an intrinsic combinatorial definition)

Zorich cocycle Lyapunov exponents ofteichmüller flows p 47/6 Fix some (irreducible) Rauzy class and consider Previously, we introduced the Zorich renormalization and the Zorich cocycle for translation surfaces, with Dropping, we find the Zorich renormalization and Zorich cocycle for interval exchange maps For our purposes, is equivalent to the inverse limit of

Checking the hypotheses # # Lyapunov exponents ofteichmüller flows p 48/6 # is a Markov map: There exists a finite partition and a countable partition such that for every

Checking the hypotheses # # Lyapunov exponents ofteichmüller flows p 48/6 # is a Markov map: There exists a finite partition and a countable partition such that for every There exists a -invariant probability which is ergodic and equivalent to volume Moreover, has bounded distortion

Zorich cocycle Lyapunov exponents ofteichmüller flows p 49/6 There is an -invariant subbundle the vector bundle, such that the fibers have dimension permutation pair: Lyapunov exponents of the cocycle transverse to of and they depend only on the vanish (in a trivial fashion) in the directions restricted Zorich cocycle preserves a symplectic form on this subbundle Thus, the Lyapunov spectrum of the Zorich cocycle has the form

Zorich cocycles are simple Lyapunov exponents ofteichmüller flows p 50/6 The goal is to prove that the are all distinct and positive By Theorem 1 and previous observations, all we need is Theorem 2 Every restricted Zorich cocycle is twisting and pinching The proof is by induction on the complexity ( number of singularities, genus of the surface) of the stratum: we look for orbits of that spend a long time close to the boundary of each stratum (hence, close to simpler strata)

Hierarchy of strata Lyapunov exponents ofteichmüller flows p 51/6 Let be any stratum Collapsing two or more singularities of an together (multiplicities add) one obtains an Abelian differential in another stratum, on the boundary of rag replacements 6 We write be attained from if can by collapsing singularities but

Starting the induction % Lyapunov exponents ofteichmüller flows p 52/6 The case,, is easy: There is only one permutation pair Top case is and bottom case is, with and

Starting the induction % % Lyapunov exponents ofteichmüller flows p 52/6 The case,, is easy: There is only one permutation pair Top case is and bottom case is, with and is hyperbolic and so its powers have arbitrarily large eccentricity This proves pinching

Starting the induction % % % Lyapunov exponents ofteichmüller flows p 53/6 To prove twisting, consider and * + PSfrag replacements Fix large enough so that no the eigenspaces of Then, that is, for all and any sufficiently large coincides with any of

# Inductive step # # Lyapunov exponents ofteichmüller flows p 54/6 Fix any pair and denote by corresponding to orbit segments that the submonoid of such It suffices to prove that the action of on the space is simple The proof is by induction on the complexity of the stratum, with respect to the order and to the genus This is easier to implement at the level of interval exchange transformations In that setting, approaching the boundary corresponds to making some coefficient very small Then it remains small for a long time, under iteration by the renormalization operator

Reduction and extension 6 6 Lyapunov exponents ofteichmüller flows p 55/6 We consider operations of simple reduction/extension: 6 0 6 6 6 0 6 6 0 6 6 6 0 6 extension: inserted letter can not be last in either row and can not be first in both rows simultaneously Lemma Given any Moreover, either there exists such that or is a simple extension of

Proof of Theorem 2 Lyapunov exponents ofteichmüller flows p 56/6 For proving twsting and pinching, we take advantage of the symplectic structure: Proposition 3 The action of twists isotropic subspaces of That is, the twisting property holds for the subspaces which the symplectic form vanishes identically on To compensate for this weaker twisting statement, we prove a stronger form of pinching:

Proof of Theorem 2 & Lyapunov exponents ofteichmüller flows p 57/6 Proposition 4 The action of That is, given any which and on there exist is strongly pinching for all in the monoid for For symplectic actions in, twisting = isotropic twisting and pinching = strong pinching In any dimension, Lemma isotropic twisting & strong pinching twisting & pinching This reduces Theorem 2 to proving the two propositions

Proof of Proposition 3 Lyapunov exponents ofteichmüller flows p 58/6 Take such that is a simple extension of If action of then there is a symplectic isomorphism that conjugates the action of on to the on Therefore, by induction, does twist isotropic subspaces

Proof of Proposition 3 Lyapunov exponents ofteichmüller flows p 59/6 If, there is some symplectic reduction of some symplectic isomorphism and that conjugates the action of on to the action induced by on symplectic reduction: quotient by of the symplectic orthogonal of, where induced action: the action on of the stabilizer of, that is, the submonoid of elements of that preserve Lemma If the action of on twists isotropic subspaces, then is minimal and its induced action on twists isotropic subspaces of The action of on the projective space minimal, that is, all its orbits are dense is indeed

References Lyapunov exponents ofteichmüller flows p 60/6 A Avila, M Viana, Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture Preprint (wwwpreprintimpabr) 2005 A Avila, M Viana, Simplicity of Lyapunov spectra: a sufficient criterion Preprint (wwwpreprintimpabr) 2006 C Bonatti, M Viana, Lyapunov exponents with multiplicity for deterministic products of matrices Ergod Th & Dynam Syst 24 (2004), 1295 1330 G Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus Ann of Math 155 (2002), 1 103 I Ya Gol dsheid, G A Margulis, Lyapunov exponents of a product of random matrices Uspekhi Mat Nauk 44 (1989), 13 60; translation in Russian Math Surveys 44 (1989), 11 71 Y Guivarc h, A Raugi, Products of random matrices: convergence theorems Random matrices and their applications, Contemp Math 50 (1986), 31 54 S Kerckhoff, H Masur, J Smillie, Ergodicity of billiard flows and quadratic differentials Ann of Math 124(1986), 293 311

References Lyapunov exponents ofteichmüller flows p 61/6 M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities Invent Math 153 (2003), 631 678 F Ledrappier, Positivity of the exponent for stationary sequences of matrices Lyapunov exponents, Lecture Notes in Math 1186 (1986), 56 73 H Masur, Interval exchange transformations and measured foliations Ann of Math 115 (1982), 169 200 G Rauzy, Echanges d intervalles et transformations induites Acta Arith 34 (1979), 315 328 S Schwartzman, Asymptotic cycles Ann of Math 66 (1957), 270 284 W Veech, Gauss measures for transformations on the space of interval exchange maps Ann of Math 115 (1982), 201 242 A Zorich, Finite Gauss measure on the space of interval exchange transformations Lyapunov exponents Ann Inst Fourier 46 (1996), 325 370 A Zorich, How do the leaves of a closed -form wind around a surface? Pseudoperiodic topology, Amer Math Soc Transl Ser 2, 197 (1999), 135 178