Lecture 1: Derivatives

Transcription

1 Lecture 1: Derivatives Steven Hurder University of Illinois at Chicago hurder/talks/ Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

2 Some basic examples Many talks on with foliations in the title start with this example, the 2-torus foliated by lines of irrational slope: Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

3 Some basic examples Many talks on with foliations in the title start with this example, the 2-torus foliated by lines of irrational slope: Never trust a talk which starts with this example! It is just too simple. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

4 Some basic examples Although, the example can be salvaged, by considering that the same example might have leaves that look like this: Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

5 Some basic examples Although, the example can be salvaged, by considering that the same example might have leaves that look like this: The suspension construction an its generalizations are very useful for producing examples. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

6 Some basic examples, 2 More interesting are talks which discuss more irregular flows such as this: Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

7 Some basic examples, 2 More interesting are talks which discuss more irregular flows such as this: Every orbit limits into the circle, so at least things have a direction. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

8 Some basic examples, 3 Earnest foliation talks start with this example, immortalized by Reeb: Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

9 Some basic examples, 3 Earnest foliation talks start with this example, immortalized by Reeb: Now begins the real questions what does it mean to discuss foliation dynamics? What is dynamic about this example? Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

10 Foliation dynamics Study the asymptotic properties of leaves of F - What is the topological shape of minimal sets? Invariant measures: can you quantify their rates of recurrence? Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

11 Foliation dynamics Study the asymptotic properties of leaves of F - What is the topological shape of minimal sets? Invariant measures: can you quantify their rates of recurrence? Directions of stability and instability of leaves - Exponents: are there directions of exponential divergence? Stable manifolds: dynamically defined transverse invariant manifolds? Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

12 Foliation dynamics Study the asymptotic properties of leaves of F - What is the topological shape of minimal sets? Invariant measures: can you quantify their rates of recurrence? Directions of stability and instability of leaves - Exponents: are there directions of exponential divergence? Stable manifolds: dynamically defined transverse invariant manifolds? Quantifying chaos - Define a measure of transverse chaos foliation entropy Estimate the entropy using linear approximations Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

13 Foliation dynamics Study the asymptotic properties of leaves of F - What is the topological shape of minimal sets? Invariant measures: can you quantify their rates of recurrence? Directions of stability and instability of leaves - Exponents: are there directions of exponential divergence? Stable manifolds: dynamically defined transverse invariant manifolds? Quantifying chaos - Define a measure of transverse chaos foliation entropy Estimate the entropy using linear approximations Shape of minimal sets - Hyperbolic exotic minimal sets Distal exceptional minimal sets Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

14 First definitions M is a compact Riemannian manifold without boundary. F is a codimension q-foliation, transversally C r for r [1, ). Transition functions for the foliation charts ϕ i : U i [ 1, 1] n T i are C leafwise, and vary C r with the transverse parameter: Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

15 Holonomy - flows Recall for a flow ϕ t : M M the orbits define 1-dimensional leaves of F. Choose a cross-section N M which is transversal to the orbits, and intersects each orbit (so N need not be connected) then for each x T there is some least τ x > 0 so that ϕ τx (x) N the return time for x. The induced map f (x) = ϕ τx (x) is a Borel map f : N N the holonomy of the flow. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

16 Holonomy - foliations Let L w be leaf of F containing w no such concept as future or past. Rather, choose z L x and smooth path τ w,z : [0, 1] L w. Cover path τ w,z by foliation charts and slide open subset U w of transverse disk S w along path to open subset W z of transverse disk S z Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

17 Holonomy pseudogroup Standardize above by choosing finite covering of M by foliation charts, with transversal sections T = T 1 T k M. The holonomy of F defines pseudogroup G F on T which is compactly generated in sense of Haefliger. Given w T, z L w T and path τ w,z : [0, 1] L w from w to z, we obtain h τw,z : U w W z where now *) h τw,z depends on the leafwise homotopy class of the path *) maximal sizes of the domain U w and range W z depends on τ w,z *) {h τw,z : U w W z } generates G F. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

18 Holonomy pseudogroup Standardize above by choosing finite covering of M by foliation charts, with transversal sections T = T 1 T k M. The holonomy of F defines pseudogroup G F on T which is compactly generated in sense of Haefliger. Given w T, z L w T and path τ w,z : [0, 1] L w from w to z, we obtain h τw,z : U w W z where now *) h τw,z depends on the leafwise homotopy class of the path *) maximal sizes of the domain U w and range W z depends on τ w,z *) {h τw,z : U w W z } generates G F. Proposition: We can assume τ w,z is a leafwise geodesic path. Proof: Each leaf L w is complete for the induced metric. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

19 Transverse differentials Let ϕ: R M M be a smooth non-singular flow for vector field X. Defines foliation F. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

20 Transverse differentials Let ϕ: R M M be a smooth non-singular flow for vector field X. Defines foliation F. For w = ϕ t (w), the Jacobian matrix Dϕ t : T w T z M. Group Law ϕ s ϕ t = ϕ s+t = Dϕ s ( X w ) = X z Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

21 Transverse differentials Let ϕ: R M M be a smooth non-singular flow for vector field X. Defines foliation F. For w = ϕ t (w), the Jacobian matrix Dϕ t : T w T z M. Group Law ϕ s ϕ t = ϕ s+t = Dϕ s ( X w ) = X z (... boring!) Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

22 Transverse differentials Let ϕ: R M M be a smooth non-singular flow for vector field X. Defines foliation F. For w = ϕ t (w), the Jacobian matrix Dϕ t : T w T z M. Group Law ϕ s ϕ t = ϕ s+t = Dϕ s ( X w ) = X z (... boring!) Normal bundle to flow Q = TM/ X = TM/T F T F. Riemannian metric on TM induces metrics on Q w for all w M. Measure for norms of maps Dϕ t : Q w Q z. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

23 Un poquito de Pesin Theory Definition: w M is hyperbolic point of flow if e F (w) lim T sup s T { 1 s log{ (Dϕ t : Q w Q z ) ± } s t s} > 0 Lemma: Set of hyperbolic points H(ϕ) = {w M e F (w) > 0} is flow-invariant. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

24 Un poquito de Pesin Theory Definition: w M is hyperbolic point of flow if e F (w) lim T sup s T { 1 s log{ (Dϕ t : Q w Q z ) ± } s t s} > 0 Lemma: Set of hyperbolic points H(ϕ) = {w M e F (w) > 0} is flow-invariant. Pesin Theory of C 2 -flows studies properties of the set of hyperbolic points. Proposition: Closure H(ϕ) M contains an invariant ergodic probability measure µ for ϕ, for which there exists λ > 0 such that for µ -a.e. w, e F (w) = lim T { 1 T log{ Dϕ T : Q w Q z } = λ Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

25 Un poquito de Pesin Theory Definition: w M is hyperbolic point of flow if e F (w) lim T sup s T { 1 s log{ (Dϕ t : Q w Q z ) ± } s t s} > 0 Lemma: Set of hyperbolic points H(ϕ) = {w M e F (w) > 0} is flow-invariant. Pesin Theory of C 2 -flows studies properties of the set of hyperbolic points. Proposition: Closure H(ϕ) M contains an invariant ergodic probability measure µ for ϕ, for which there exists λ > 0 such that for µ -a.e. w, e F (w) = lim T { 1 T log{ Dϕ T : Q w Q z } = λ Proof: Just calculus! (plus usual subadditive tricks) Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

26 Foliation geodesic flow Let w M and consider L w as complete Riemannian manifold. For v T w F = T w L w with v w = 1, there is unique geodesic τ w, v (t) starting at w with τ w, v (0) = v Define ϕ w, v : R M, ϕ w, v (w) = τ w, v (t) Let M = T 1 F denote the unit tangent bundle to the leaves, then we obtain the foliation geodesic flow ϕ F t : R M M Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

27 Foliation geodesic flow Let w M and consider L w as complete Riemannian manifold. For v T w F = T w L w with v w = 1, there is unique geodesic τ w, v (t) starting at w with τ w, v (0) = v Define ϕ w, v : R M, ϕ w, v (w) = τ w, v (t) Let M = T 1 F denote the unit tangent bundle to the leaves, then we obtain the foliation geodesic flow ϕ F t : R M M Remark: ϕ F t preserves the leaves of the foliation F on M whose leaves are the unit tangent bundles to leaves of F. = Dϕ F t preserves the normal bundle Q M for F. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

28 Foliation exponents Definitions: (H) ŵ M is hyperbolic if e F (ŵ) lim T sup s T { 1 s log{ (DϕF t : Qŵ Qẑ) ± } s t s} > 0 Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

29 Foliation exponents Definitions: (H) ŵ M is hyperbolic if e F (ŵ) lim T sup s T { 1 s log{ (DϕF t : Qŵ Qẑ) ± } s t s} > 0 (E) ŵ M is elliptic if e F (ŵ) = 0, and there exists κ(ŵ) such that { (Dϕ F t : Qŵ Qẑ) ± κ(ŵ) for all t R Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

30 Foliation exponents Definitions: (H) ŵ M is hyperbolic if e F (ŵ) lim T sup s T { 1 s log{ (DϕF t : Qŵ Qẑ) ± } s t s} > 0 (E) ŵ M is elliptic if e F (ŵ) = 0, and there exists κ(ŵ) such that { (Dϕ F t : Qŵ Qẑ) ± κ(ŵ) for all t R (P) ŵ M is parabolic if e F (ŵ) = 0, and ŵ is not elliptic. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

31 Dynamical decomposition of foliations Theorem: Let F be a C 1 -foliation of a compact Riemannian manifold M. Then there exists a decomposition of M into F-saturated Borel subsets M = M H M P M E where the derivative for the geodesic flow of F satisfies: Dϕ F t is transversally hyperbolic for L w M H Dϕ F t is bounded (in time) for L w M E Dϕ F t has subexponential growth (in time), but is not bounded, for L w M P Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

32 Transversally hyperbolic measures Definition: An invariant probability measure µ for the foliation geodesic flow on M is said to be transversally hyperbolic if e F (ŵ) = λ > 0 for µ -a.e. ŵ. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

33 Transversally hyperbolic measures Definition: An invariant probability measure µ for the foliation geodesic flow on M is said to be transversally hyperbolic if e F (ŵ) = λ > 0 for µ -a.e. ŵ. Theorem: Let F be a C 1 foliation of a compact manifold. If M H, then the foliation geodesic flow has at least one transversally hyperbolic ergodic measure. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

34 Transversally hyperbolic measures Definition: An invariant probability measure µ for the foliation geodesic flow on M is said to be transversally hyperbolic if e F (ŵ) = λ > 0 for µ -a.e. ŵ. Theorem: Let F be a C 1 foliation of a compact manifold. If M H, then the foliation geodesic flow has at least one transversally hyperbolic ergodic measure. Proof: The proof is technical, but is actually just calculus applied to the foliation pseudogroup. Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

35 Standard examples, revisited: 1 For the linear foliation, every point is elliptic (it is Riemannian!) However, if F is a C 1 -foliation which is topologically semi-conjugate to a linear foliation, so is a generalized Denjoy example, then M = M P! Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

36 Standard examples, revisited: 2 The case of the Reeb foliation on the solid torus is more interesting: Pick w M and a direction, v T w L w, then follow the geodesic τ w, w (t). It is asymptotic to the boundary torus, so defines a limiting Schwartzman cycle on the torus for some flow. Thus, it limits on either a circle, or a lamination. This will be a hyperbolic measure if the holonomy of the compact leaf is hyperbolic. The exponent depends on the direction! Steven Hurder (UIC) Dynamics of Foliations May 3, / 19

37 References S. Hurder, Ergodic theory of foliations and a theorem of Sacksteder, in Dynamical Systems: Proceedings, University of Maryland Lect. Notes in Math. Vol. 1342, pages , P. Walczak, Dynamics of the geodesic flow of a foliation, Ergodic Theory Dynamical Systems, 8: , L. Barreira and Ya.B. Pesin, Lyapunov exponents and smooth ergodic theory, University Lecture Series, Vol. 23, AMS, Steven Hurder (UIC) Dynamics of Foliations May 3, / 19