Perspectives and Problems on Mapping Class Groups III: Dynamics of pseudo-anosov Homeomorphisms

Transcription

1 Perspectives and Problems on Mapping Class Groups III: Dynamics of pseudo-anosov Homeomorphisms Dan Margalit Georgia Institute of Technology IRMA, June 2012

2 The Nielsen Thurston Classification Theorem (Nielsen, Thurston, Fathi Laudenbach Poénaru). Each f Mod(S g ) has a representative φ of one of the following types: Periodic: φ n = 1 Reducible: φ(m) = M for some 1-submanifold M Pseudo-Anosov: There are two transverse, singular, arational measured foliations and a stretch factor λ so that φ stretches along one foliation by λ and the other by 1/λ Compare with the action of SL(2, Z) on T 2 (or R 2 ).

3 Properties of the stretch factor λ logλ = growth rate of lengths of curves: n l(f n (c)) = λ lim n log λ = growth rate of intersection numbers: n i(b,f n (c)) = λ lim n log λ = topological entropy of f log λ = length of the loop [f ] in moduli space

4 Pseudo-Anosovs via Train tracks Question. How to find a single pseudo-anosov f? Penner 88: Let φ Homeo + (S). Suppose filling, birecurrent train track τ where τ splits to φ(τ). τ φ(τ) The transition matrix M is PF with PF eigenvalue λ. Then [φ] is pseudo-anosov with stretch factor λ.

5 A Taffy Puller Example

6 The Thurston Sullivan Mural 1971

7 The Taffy Puller Example

8 The Taffy Puller Example

9 The Taffy Puller Example

10 The Taffy Puller Example

11 The Taffy Puller Example

12 The Taffy Puller Example

13 The Taffy Puller Example

14 The Taffy Puller Example y x w x y=z w z

15 The Taffy Puller Example z w y x

16 The Taffy Puller Example

17 The Taffy Puller Example

18 The Taffy Puller Example

19 The Taffy Puller Example 4x+z+2w= 2x+2y+3z y+2z x+2y+4z

20 The Taffy Puller Example 4x+z+2w= 2x+2y+3z y+2z x+2y+4z λ = (1 + 2) 2

21 Pseudo-Anosovs via Train tracks Train track to Foliation

22 Pseudo-Anosovs via Train tracks Train track to Foliation

23 Pseudo-Anosovs via Train tracks Train track to Foliation

24 Penner s Construction Theorem (Penner 88). Let {a i } and {b i } be two multicurves that fill S. Any product of the T ai and T 1 b j where each a i and b j occur at least once is pseudo-anosov. Conjecture (Penner). Each pseudo-anosov mapping class has a power arising from the above construction.

25 Penner s Construction Theorem (Penner 88). Let {a i } and {b i } be two multicurves that fill S. Any product of the T ai and T 1 b j where each a i and b j occur at least once is pseudo-anosov. Conjecture (Penner). Each pseudo-anosov mapping class has a power arising from the above construction.

26 Penner s Construction Theorem (Penner 88). Let {a i } and {b i } be two multicurves that fill S. Any product of the T ai and T 1 b j where each a i and b j occur at least once is pseudo-anosov. Conjecture (Penner). Each pseudo-anosov mapping class has a power arising from the above construction.

27 Penner s Construction Theorem (Penner 88). Let {a i } and {b i } be two multicurves that fill S. Any product of the T ai and T 1 b j where each a i and b j occur at least once is pseudo-anosov. Conjecture (Penner). Each pseudo-anosov mapping class has a power arising from the above construction.

28 Penner s Construction Theorem (Penner 88). Let {a i } and {b i } be two multicurves that fill S. Any product of the T ai and T 1 b j where each a i and b j occur at least once is pseudo-anosov. Conjecture (Penner). Each pseudo-anosov mapping class has a power arising from the above construction.

29 The Bestvina Handel Algorithm Theorem (Bestvina Handel 92) Explicit algorithm to find the train track and determine the Nielsen Thurston type of any mapping class. Problem Determine the complexity of the Bestvina Handel algorithm.

30 Number theory of the Stretch Factor

31 Degree restrictions Theorem (Thurston 76). For pseudo-anosov f Mod(S g ), the stretch factor λ(f ) is an algebraic integer of degree at most 6g 6. Proof (McMullen). Consider the action on train track coordinates. Theorem (Long 85). For pseudo-anosov f Mod(S g ), if the algebraic degree of the stretch factor λ(f ) is greater than 3g 3, then it is even. Proof (McMullen). The characteristic polynomial P for f has degree at most 6g 6. It is symplectic, so roots come in reciprocal pairs. If the minimal polynomial for λ is not even, then it is not reciprocal, so there is a another factor of the same degree in P.

32 Two Conjectures Folk Conjecture. A real number λ > 1 is a pseudo-anosov stretch factor if and only if λ and λ 1 are both Perron numbers. (Lind 84: A Perron number is a PF eigenvalue of a PF matrix.) Conjecture. For any 2 k 3g 3 or any 3g 3 k 6g 6 even, there exists pseudo-anosov f Mod(S g ) whose stretch factor has algebraic degree k. Arnoux Yoccoz 81. There are pas in Mod(S g ) of degree g. Shin 12. The conjecture is true for 1 g 3. Claim (Thurston). The Thurston construction of pseudo-anosov mapping classes in Mod(S g ) gives stretch factors of degree 6g 6.

33 Small stretch factors

34 Small stretch factors Theorem (Arnoux Yoccoz 81, Ivanov 88). For fixed S, the set {λ(f ) f Mod(S) pseudo-anosov} is closed and discrete in R least element MinStretch(S). ( MinStretch(T 2 ) ) = , realized by ( ) MinStretch(S 2 ) = largest real root(x 4 x 3 x 2 + 1) (Zhirov 95, Cho-Ham 08) MinStretch(S g ) unknown for g 3. Problem. Understand the maps with smallest stretch factors: In which subgroups do they (not) lie? What do their foliations look like? Which 3 manifolds result?

35 Asymptotics of MinStretch(S g ) S g = closed surface of genus g Theorem (Penner). There are constants m,m so that m log(minstretch(s g )) χ(s g ) M In other words: log(minstretch(s g )) 1 g So, as g, there are stretch factors arbitrarily close to 1.

36 Penner s Small-Stretch Pseudo-Anosovs ρ σ φ=ρσ

37 Asymptotics of MinStretch(S g ) Question. Is MinStretch(S g ) monotone in g? Question. Compute the limit if it exists: Does it equal MinStretch(S 1 )? lim MinStretch(S g) g g

38 Symmetry conjecture Conjecture (Farb Leininger Margalit). If log(λ(f )) is small enough, say M/g, then f is the product of a periodic mapping class with a reducible mapping class, supported on a subsurface of uniformly small complexity (depending on M).

39 Algebraic complexity implies dynamical complexity

40 Trivial action on homology implies large stretch Thurston: pseudo-anosov maps in I(S g ). Theorem (Farb Leininger Margalit). Let g 2. If f I(S g ), then λ(f ) > The bound does not depend on g.

41 Algebraic complexity implies dynamical complexity Let {N k (S g )} denote the Johnson filtration of Mod(S g ). Theorem (Farb Leininger Margalit). There exist constants m k with lim k m k = so that if f N k (S g ) is pseudo-anosov, then λ(f ) > m k. The constants m k are universal: they do not depend on g.

42 Algebraic complexity implies dynamical complexity Idea of Proof Theorem. If f I(S g ) is pseudo-anosov, then λ(f ) > (independent of g). Ingredient 1. For all simple closed curves c, we have i(c,f k (c)) 4 for some k {1,2,3,4}. Ingredient 2. There exists a metric on S g so that l(f (c)) l(c) < λ(f ) for all simple closed curves c.

43 Smallest Stretch Factors for Subgroups For G < Mod(S g ), define MinStretch(G) = min{λ(f ) : f G pseudo-anosov} Above: lim k MinStretch(N k(s g )) =. Question. Is MinStretch(N k (S g )) monotone in k? Is MinStretch(I(S g )) < MinStretch(K(S g ))? Let I k (S g ) = the subgroup of Mod(S g ) fixing a k-dimensional subspace of H 1 (S g ; Z). Question (Ellenberg). Are there constants m and M so that m(k + 1) g log(minstretch(i k (S g ))) M(k + 1)? g

44 Geometric complexity implies dynamical complexity

45 The mapping torus construction Given φ Homeo + (S), we obtain a 3 manifold M φ = called the mapping torus of φ. S [0,1] (x,0) (φ(x),1) Theorem (Thurston). M φ hyperbolic [φ] pseudo-anosov

46 Thurston s Fibering Theorem and the McMullen Construction Consider M φ, where φ is pseudo-anosov. Assume that φ fixes some nontrivial x H 1 (S, Z). Theorem (Thurston). The mapping torus M φ fibers in infinitely many ways, finitely many with fiber of fixed genus g. infinitely many φ i Homeo(S i ) with genus(s i ), all with M φi M φ McMullen: There is a sequence {φ g } that satisfies Penner s asymptotics: log(λ((φ g )) 1 g

47 The set of all small-stretch pseudo-anosov maps Consider the set of all small-stretch pseudo-anosovs of all surfaces: SmallStretch P = {φ : S S χ(s) < 0, φ pseudo-anosov, log(λ((φ))) χ(s) < P} Penner For P large enough, SmallStretch P is infinite. Consider the a priori infinite set of mapping tori: {M φ φ SmallStretch P } A bold guess: this set is finite. Not true! (Penner s examples)

48 Finitely many mapping tori Next try: Given φ : S S pseudo-anosov, let S = S {singularities of foliations for φ}. There is an induced pseudo-anosov map: φ S : S S Theorem (Farb Leininger Margalit 09). For any P, the set is finite. {M φ S φ SmallStretch P }

49 Finitely many mapping tori Theorem (Farb Leininger Margalit). For any P, the set is finite. Ω P = {M φ S φ SmallStretch P } Question. What is Ω P? Question. What is the smallest number of mapping tori needed to generate all of the MinStretch(S g ) maps? Is it two?

50 Main tool: Markov partitions

51 Geometric complexity implies dynamical complexity Idea of the proof Theorem (Farb Leininger Margalit). For any P, the set {M φ S φ SmallStretch P } is finite. Step 1. Given φ SmallStretch P, find a small Markov partition. Step 2. Show that all but f (P) rectangles get mapped homeomorphically onto other rectangles. Step 3. In the mapping torus, crush the corresponding prisms. Agol 10: Train track proof

52 Geometric complexity implies dynamical complexity Idea of the proof Theorem (Farb Leininger Margalit). For any P, the set {M φ S φ SmallStretch P } is finite. Step 1. Given φ SmallStretch P, find a small Markov partition. Step 2. Show that all but f (P) rectangles get mapped homeomorphically onto other rectangles. Step 3. In the mapping torus, crush the corresponding prisms. Agol 10: Train track proof

53 Geometric complexity implies dynamical complexity Idea of the proof Theorem (Farb Leininger Margalit). For any P, the set {M φ S φ SmallStretch P } is finite. Step 1. Given φ SmallStretch P, find a small Markov partition. Step 2. Show that all but f (P) rectangles get mapped homeomorphically onto other rectangles. Step 3. In the mapping torus, crush the corresponding prisms. Agol 10: Train track proof

54 Geometric complexity implies dynamical complexity Idea of the proof Theorem (Farb Leininger Margalit). For any P, the set {M φ S φ SmallStretch P } is finite. Step 1. Given φ SmallStretch P, find a small Markov partition. Step 2. Show that all but f (P) rectangles get mapped homeomorphically onto other rectangles. Step 3. In the mapping torus, crush the corresponding prisms. Agol 10: Train track proof

55 Geometric complexity implies dynamical complexity Idea of the proof Theorem (Farb Leininger Margalit). For any P, the set {M φ S φ SmallStretch P } is finite. Step 1. Given φ SmallStretch P, find a small Markov partition. Step 2. Show that all but f (P) rectangles get mapped homeomorphically onto other rectangles. Step 3. In the mapping torus, crush the corresponding prisms. Agol 10: Train track proof

56 Geometric complexity implies dynamical complexity Idea of the proof Theorem (Farb Leininger Margalit). For any P, the set {M φ S φ SmallStretch P } is finite. Step 1. Given φ SmallStretch P, find a small Markov partition. Step 2. Show that all but f (P) rectangles get mapped homeomorphically onto other rectangles. Step 3. In the mapping torus, crush the corresponding prisms. Want to crush enough to get finiteness, but not so much that the topology changes. Agol 10: Train track proof