The Zorich Kontsevich Conjecture

Transcription

1 The Zorich Kontsevich Conjecture Marcelo Viana (joint with Artur Avila) IMPA - Rio de Janeiro The Zorich Kontsevich Conjecture p.1/27

2 Translation Surfaces Compact Riemann surface endowed with a non-vanishing holomorphic closed -form (Abelian differential). The Zorich Kontsevich Conjecture p.2/27

3 Translation Surfaces Compact Riemann surface endowed with a non-vanishing holomorphic closed -form (Abelian differential). Compact orientable surface endowed with a flat metric with finitely many conical singularities and a unit parallel vector field. The Zorich Kontsevich Conjecture p.2/27

4 Translation Surfaces Compact Riemann surface endowed with a non-vanishing holomorphic closed -form (Abelian differential). singularities = zeros of such that Compact orientable surface endowed with a flat metric with finitely many conical singularities and a unit parallel vector field. The Zorich Kontsevich Conjecture p.2/27

5 Construction of Translation Surfaces Consider a planar polygon whose sides can be grouped in pairs of (non-adjacent) segments that are parallel and have the same length. S E W N Identifying the two sides in the same pair, by translation, one obtains a translation surface. Euclidean singularities vertices The Zorich Kontsevich Conjecture p.3/27

6 Geodesic Flows on Translation Surfaces 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? The Zorich Kontsevich Conjecture p.4/27

7 Measured Foliations A foliation on a surface is measured if it is defined by some -form with isolated zeros: the leaves are tangent to the kernel of at every non-singular point. Example: parallel foliations on translation surfaces. singularities vertices The Zorich Kontsevich Conjecture p.5/27

8 Measured Foliations Maier 43: Given a measured foliation, the ambient splits into periodic components: all leaves are closed; minimal components: all leaves are dense; separated by saddle-connections or homoclinic loops. The Zorich Kontsevich Conjecture p.6/27

9 Measured Foliations Maier 43: Given a measured foliation, the ambient splits into periodic components: all leaves are closed; minimal components: all leaves are dense; separated by saddle-connections or homoclinic loops. Calabi 69, Katok 73: Every measured foliation without saddle-connections is the vertical foliation with respect to some flat translation metric. Necessary and sufficient: no closed paths homologous to zero formed by positively oriented saddle-connections. The Zorich Kontsevich Conjecture p.6/27

10 Interval Exchange Transformations Associated to the vertical foliation on a translation surface there is an interval exchange transformation: the return map to some section transverse to the foliation. PSfrag replacements The Zorich Kontsevich Conjecture p.7/27

11 Interval Exchange Transformations Associated to the vertical foliation on a translation surface there is an interval exchange transformation: the return map to some section transverse to the foliation. PSfrag replacements Conversely, every interval exchange transformation may be suspended to the vertical flow on some translation surface (not unique). The Zorich Kontsevich Conjecture p.7/27

12 Geodesic Flows on the Flat Torus V H Let define the direction of the geodesics. If is rational then every geodesic is closed. If is irrational then the flow is uniquely ergodic. The Zorich Kontsevich Conjecture p.8/27

13 Geodesic Flows on the Flat Torus PSfrag replacements Let define the direction of the geodesics. If is rational then every geodesic is closed. If is irrational then the flow is uniquely ergodic. Given a geodesic segment of length. Then, close it to get is bounded. The Zorich Kontsevich Conjecture p.9/27

14 Geodesic Flows in Higher Genus placements, close it to get an. Given any geodesic segment of length element of The Zorich Kontsevich Conjecture p.10/27

15 Asymptotic Cycles Schwartzmann 57: The asymptotic cycle of a pair (surface, direction) is the limit Kerckhoff, Masur, Smillie 86: For every translation surface and for almost every direction, the geodesic flow is uniquely ergodic. In particular, the asymptotic cycle is well defined, and every geodesic is dense. The Zorich Kontsevich Conjecture p.11/27

16 Deviations from the Limit Direction Zorich discovered that the deviation of the vector from the direction of distributes itself along a favorite direction, with amplitude for some : Sfrag replacements The Zorich Kontsevich Conjecture p.12/27

17 Deviations from the Limit Direction The same phenomenon is observed in higher orders: the component of orthogonal to has a favorite direction, and amplitude for some, and so on. ag replacements The Zorich Kontsevich Conjecture p.13/27

18 Conjecture and Main Result Conjecture (Zorich, Kontsevich). There are numbers and subspaces with and for every, such that the deviation of from has amplitude for all the deviation of from (where = genus of the flat surface). is bounded. The Zorich Kontsevich Conjecture p.14/27

19 Conjecture and Main Result Conjecture (Zorich, Kontsevich). There are numbers and subspaces with and for every, such that the deviation of from has amplitude for all the deviation of from (where = genus of the flat surface). is bounded. Theorem (Avila, Viana 04). The Z-K conjecture is true. There was fundamental previous work by Forni 02. Our argument contains another proof of Forni s result. The Zorich Kontsevich Conjecture p.14/27

20 Conjecture and Main Result Conjecture (Zorich, Kontsevich). There are numbers and subspaces with and for every, such that the deviation of from has amplitude for all the deviation of from (where = genus of the flat surface). is bounded. Theorem (Avila, Viana 04). The Z-K conjecture is true. There was fundamental previous work by Forni 02. Our argument contains another proof of Forni s result Work of Kontsevich and Zorich translated the claim of the conjecture into a statement in Dynamics:. The Zorich Kontsevich Conjecture p.14/27

21 The Rauzy Algorithm To analyze the behavior of longer and longer geodesics, we consider return maps to shorter and shorter cross-sections. One way to do this is the Rauzy renormalization algorithm in the space of interval exchange transformations. We describe the algorithm through an example. The Zorich Kontsevich Conjecture p.15/27

22 ements The Rauzy Algorithm Each interval exchange transformation is replaced by the corresponding return map to a certain subinterval. Above is a bottom case: of the two rightmost intervals, the bottom one is longest. The Zorich Kontsevich Conjecture p.16/27

23 The Rauzy Algorithm ements The Rauzy transformation is defined by. It admits an invariant measure absolutely continuous with respect to Lebesgue measure in the -space. The Zorich Kontsevich Conjecture p.17/27

24 The Rauzy Cocycle Now we analyze the effect of this algorithm on the return map (suspension of the interval exchange transformation): replacements Consider a geodesic segment that leaves from the th interval and returns to the cross-section. Close it by joining the endpoints to some chosen point. This defines some. The Zorich Kontsevich Conjecture p.18/27

25 The Rauzy Cocycle Now we analyze the effect of this algorithm on the return map (suspension of the interval exchange transformation): replacements This corresponds to a linear cocycle over the Rauzy map. The Zorich Kontsevich Conjecture p.19/27

26 The Zorich Cocycles The invariant measure of an accelerated algorithm is infinite... Zorich introduced and where is smallest such that the Rauzy iteration changes from top to bottom or vice-versa. The Zorich Kontsevich Conjecture p.20/27

27 The Zorich Cocycles The invariant measure of an accelerated algorithm is infinite... Zorich introduced and where is smallest such that the Rauzy iteration changes from top to bottom or vice-versa. The transformation admits an absolutely continuous invariant ergodic probability over every Rauzy class = smallest invariant subset of the set of permutations the cocycle acts symplectically on. This brings us to the setting of the Oseledets theorem: The Zorich Kontsevich Conjecture p.20/27

28 Conjecture and Main Result Conjecture (Zorich, Kontsevich). The Lyapunov exponents of every Zorich cocycle are non-zero and distinct: (then the asymptotic flag corresponds to the Oseledets decomposition). The Zorich Kontsevich Conjecture p.21/27

29 Conjecture and Main Result Conjecture (Zorich, Kontsevich). The Lyapunov exponents of every Zorich cocycle are non-zero and distinct: (then the asymptotic flag corresponds to the Oseledets decomposition). Theorem (Veech 84).. Theorem (Forni 02).. The Zorich Kontsevich Conjecture p.21/27

30 Conjecture and Main Result Conjecture (Zorich, Kontsevich). The Lyapunov exponents of every Zorich cocycle are non-zero and distinct: (then the asymptotic flag corresponds to the Oseledets decomposition). Theorem (Veech 84).. Theorem (Forni 02).. Theorem (Avila, Viana 04). All exponents are non-zero and distinct. The Zorich Kontsevich Conjecture p.21/27

31 Main Steps of the Proof 1. A general criterium for multiplicity of the Lyapunov exponents of linear cocycles. 2. Checking that this criterium applies to every Zorich cocycle. Multiplicity of a Lyapunov exponent = dimension of the corresponding invariant subbundle in the Oseledets decomposition. The Zorich Kontsevich Conjecture p.22/27

32 Linear Cocycles Let They define a linear cocycle be a measurable transformation and be a measurable function., through Assume: 1. The map has a finite or countable Markov partition. 2. is constant on each element of the partition. 3. A bounded distortion condition (inverse branches of iterates of and their Jacobians are equicontinuous). The Zorich Kontsevich Conjecture p.23/27

33 The Criterium We call the cocycle simple if the map has 1. (pinching) Some periodic point such that all the eigenvalues of norms. 2. (twisting) Some homoclinic point, with period have distinct, and such that and of with for any invariant subspaces. twisting the algebraic minors of the matrix of eigenbasis of are all different from zero. in an The Zorich Kontsevich Conjecture p.24/27

34 The Criterium Theorem 0. If the cocycle is simple then all its Lyapunov exponents have multiplicity. Previous results were obtained by Guivarc h, Raugi and Gol dsheid, Margulis, for products of independent random matrices, and Bonatti, Viana, for cocycles over subshifts of finite type. The Zorich Kontsevich Conjecture p.25/27

35 Checking the Criterium Theorem 1. Every Zorich cocycle is simple. The Zorich Kontsevich Conjecture p.26/27

36 Checking the Criterium Theorem 1. Every Zorich cocycle is simple. The proof is by induction on the number of intervals. We consider combinatorial operations of reduction/extension: This has a topological and geometric counterpart for the corresponding surfaces: The Zorich Kontsevich Conjecture p.26/27

37 Checking the Criterium Given with symbols, there exists with such that is an extension of. Then, either or. symbols 1. If then the corresponding Zorich actions are (symplectically) conjugate. ng also requires a careful combinatorial analysis. The Zorich Kontsevich Conjecture p.27/27

38 Checking the Criterium Given with symbols, there exists with such that is an extension of. Then, either or. symbols 1. If then the corresponding Zorich actions are (symplectically) conjugate. 2. If then symplectic reduction of symplectic orthogonal of may be seen as a inside and the Zorich action on is conjugate to the natural action on the symplectic reduction. ng also requires a careful combinatorial analysis. The Zorich Kontsevich Conjecture p.27/27

39 Checking the Criterium Given with symbols, there exists with such that is an extension of. Then, either or. symbols 1. If then the corresponding Zorich actions are (symplectically) conjugate. 2. If then symplectic reduction of symplectic orthogonal of may be seen as a inside and the Zorich action on is conjugate to the natural action on the symplectic reduction. In this way one can prove twisting for from twisting for Pinchi ng also requires a careful combinatorial analysis.. The Zorich Kontsevich Conjecture p.27/27