Train tracks, braids, and dynamics on surfaces. Dan Margalit Georgia Tech YMC 2011

Transcription

1 Train tracks, braids, and dynamics on surfaces Dan Margalit Georgia Tech YMC 2011

2 What is topology? Movie: kisonecat (Youtube)

3 3- prong Taffy Puller Movie: coyboy6000 (Youtube)

4 4- prong Taffy Puller Movie: Amandarling8 (Youtube)

5 QuesOon Which taffy puller is more efficient the 3- prong puller or the 4- prong puller?

6 Mixing paserns as braids AnimaOon: J. Gonzàlez- Meneses

7 The braid group σ 1 (σ 2 ) 1 = σ 1 (σ 2 ) 1

8 The taffy pullers as braids (σ 1 ) 2 (σ 2 ) - 2 σ 1 σ 3 (σ 2 ) 2 σ 1 σ 3

9 The standard hair braid (σ 2 ) - 1 σ 1 Braid images made with KnotPlot

10 Adding a tracer Boyland Stremler Aref 2003

11 The standard hair braid c 2 f (c) f(c)

12 Curve to Train Track

13 The standard hair braid x+y y x x y x x y x+y 2x+y

14 Thurston- Sullivan Mural 1971 Photo: J. Behrstock

15 Thurston- Sullivan Mural 2011

16 The standard hair braid Train track Leading Eigenvalue y x = Matrix PosiOve Eigenvector (1+ 5) /2 1

17 The Stretch Factor The PF eigenvalue λ = = is the stretch factor for the standard hair braid. 2 The number of strands along any branch of the train track grows like λ n.

18 The Stretch Factor f 100 (c) has ~ 9 x horizontal strands.

19 Weighted Train Track (the PF eigenvector)

20 Train Track to FoliaOon

21 FoliaOon

22 FoliaOon to LaminaOon

23 LaminaOon

24 Theorems Thurston 1980: Every homeomorphism of a surface preserves some train track. Bestvina- Handel 1992: There is an algorithm to find the train track. Idea: Iterate on any curve!

25 The Perron- Frobenius Theorem Let M be a square matrix. Suppose M has a power where every entry is posiove. Then M has an eigenvector v (unique up to scale), with each entry posiove. The eigenvalue λ for v is posiove. v and λ are the PF eigenvector/value for M.

26 A Century of Surface Homeomorphisms JAKOB NIELSEN IN MEMORIAM BY W E R N E R FENCttEL Copenhagen Memorial address given at a meeting o f the Danish Mathematical Society on 7 December 1959 in new invariants, Was his newi t inis but naturalan that explosion the Danish Mathematical Societyknot should wish to commemorate and the open by a meeting,was for his what death meant to our meant society the loss of one question they geometrically. And variant reallyprofessor new,jakob Nielsen of its most active and prominent members. He joined it in 1921, sat on its committee for here Arnold was, with more invariants! The old or a new way to eight years, and in no less than 28 flawless and inspiring lectures given to it, he communicated to the mathematical the results of his own research,the and often alsoones that of were integers onescircle were polynomials, new look at something others. (lots of integers!). Arnold asked me to copy and disknown? He0-did Acta mathematica Imprlm6 le 29 juin 1960 tribute the paper in the United States. So one afnot know. Examternoon shortly after his arrival I made lots of phoples were needed, tocopies, and sent them out to everyone I could and a few days think of who seemed appropriate. But even as I did later we met it I suspected the knot theory community might not again, in my ofbe so overjoyed to have yet more knot invariants fice, to work some coming unexpectedly out of left field! There is reout. That was sistance to learning new things. We had just learned probably May 22, about operator algebras, and suddenly we had to Birman teaching at Columbia, The new learn about singularity theory! But Arnold kept link invariant was after me, at tea every day. a Laurent polynomial. My first thought was: it must 1987 Abe Frajnlich Xiao-Song Lin was an assistant professor in the be the Alexander polynomial. So I said, Here are department, and his field is knot theory. We ran a two knots (the trefoil and its mirror image) that have seminar together the same Alexander polynomial. Let s see if your and talked every polynomial can distinguish them. To my aston-

27 3- prong Taffy Puller 1 0

28 3- prong Taffy Puller 1 2

29 3- prong Taffy Puller 5 2

30 3- prong Taffy Puller

31 3- prong Taffy Puller " "!!

32 3- prong Taffy Puller

33 3- prong Taffy Puller

34 3- prong Taffy Puller 5x+2y 2x+y

35 3- prong Taffy Puller Train track x y PF Eigenvalue = (1+ 2) 2 Matrix PF Eigenvector

36 The Silver RaOo δ =1+ 2 δ = δ 2 2δ 1 = δ 1 = δ δ 1! P δ = lim n +1 where P 0 = 0, P 1 = 1, n P and P n n+2 = 2P n+1 + P n δ = cot π 8 0, 1, 2, 5, 12, 29, 70, 169,...

37 The Silver RaOo

38 4- prong Taffy Puller

39 4- prong Taffy Puller

40 4- prong Taffy Puller

41 4- prong Taffy Puller

42 4- prong Taffy Puller

43 4- prong Taffy Puller

44 4- prong Taffy Puller #! "! #$% " %

45 4- prong Taffy Puller z w y x

46 4- prong Taffy Puller

47 4- prong Taffy Puller

48 4- prong Taffy Puller

49 4- prong Taffy Puller y+2z 4x+z+2w= 2x+2y+3z x+2y+4z

50 4- prong Taffy Puller Train track # $ #!%" $! " PF Eigenvalue = (1+ 2) 2 Matrix PF Eigenvector

51 4- prong Taffy Puller In (x,y,z,w) coordinates, we have: n 1 = P n +2 P n +1 P n +2 P n +1 So we obtain the Pell numbers P n by applying the 4- prong taffy puller to the (1,0,1,0) curve.

52 4- prong Taffy Puller

53 Conclusion The 3- prong and 4- prong taffy pullers have the same stretch factor: ( 1+ 2) 2 In fact, the corresponding maps from the plane to itself are equivalent: by sending the second (or third) of the 4 prongs to infinity, we obtain the 3- prong map. We conclude that the 3- prong puller is more efficient.

54 Problems Exercise: Find maps that give the nth Fibonacci sequence. Open quesoon: What is the smallest stretch factor on a given surface? Open quesoon: Which real numbers are stretch factors of surfaces? Open quesoon: How hard is it to find the train tracks?

55 A Reference

56 Poincaré Recurrence Movie: InducOveload (Wikipedia)