January  2016, 36(1): 541-570. doi: 10.3934/dcds.2016.36.541

Dynnikov and train track transition matrices of pseudo-Anosov braids

1. 

Dicle University, Mathematics Department, 21280, Diyarbakır, Turkey

Received  May 2014 Revised  May 2015 Published  June 2015

We compare the spectra of Dynnikov matrices with the spectra of the train track transition matrices of a given pseudo-Anosov braid on the finitely punctured disk, and show that these matrices are isospectral up to roots of unity and zeros under some particular conditions. It is shown, via examples, that Dynnikov matrices are much easier to compute than transition matrices, and so yield data that was previously inaccessible.
Citation: S. Öykü Yurttaş. Dynnikov and train track transition matrices of pseudo-Anosov braids. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 541-570. doi: 10.3934/dcds.2016.36.541
References:
[1]

E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, 4 (1925), 47-72. doi: 10.1007/BF02950718.

[2]

E. Artin, Theory of braids, Ann. of Math. (2), 48 (1947), 101-126. doi: 10.2307/1969218.

[3]

M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology, 34 (1995), 109-140. doi: 10.1016/0040-9383(94)E0009-9.

[4]

J. Birman, P. Brinkmann and K. Kawamuro, A polynomial invariant of pseudo-Anosov maps, Journal of Topology and Analysis, 4 (2012), 13-47. doi: 10.1142/S1793525312500033.

[5]

P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest, Why are Braids Orderable?, vol. 14 of Panoramas et Synthèses [Panoramas and Syntheses], Société Mathématique de France, Paris, 2002.

[6]

P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest, Ordering Braids, vol. 148 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/surv/148.

[7]

I. Dynnikov, On a Yang-Baxter mapping and the Dehornoy ordering, Uspekhi Mat. Nauk, 57 (2002), 151-152. doi: 10.1070/RM2002v057n03ABEH000519.

[8]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston Sur les Surfaces, vol. 66 of Astérisque, Société Mathématique de France, Paris, 1979, Séminaire Orsay.

[9]

J. Franks and M. Misiurewicz, Cycles for disk homeomorphisms and thick trees, in Nielsen theory and dynamical systems (South Hadley, MA, 1992), vol. 152 of Contemp. Math., Amer. Math. Soc., Providence, RI, (1993), 69-139. doi: 10.1090/conm/152/01319.

[10]

T. Hall, Software available for download from:, , (). 

[11]

T. Hall and S. Ö. Yurttaş, On the topological entropy of families of braids, Topology Appl., 156 (2009), 1554-1564. doi: 10.1016/j.topol.2009.01.005.

[12]

J. Los, Pseudo-Anosov maps and invariant train tracks in the disc: A finite algorithm, Proc. London Math. Soc. (3), 66 (1993), 400-430. doi: 10.1112/plms/s3-66.2.400.

[13]

L. Mosher, Train track expansions of measured foliations, 2003,, Preprint available from , (). 

[14]

J.-O. Moussafir, On computing the entropy of braids, Funct. Anal. Other Math., 1 (2006), 37-46.

[15]

R. C. Penner and J. L. Harer, Combinatorics of Train Tracks, vol. 125 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1992.

[16]

E. Rykken, Expanding factors for pseudo-anosov homeomorphisms, Rocky Mountain J. Math., 28 (1998), 1103-1124. doi: 10.1216/rmjm/1181071758.

[17]

W. P. Thurston, n the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. doi: 10.1090/S0273-0979-1988-15685-6.

[18]

S. Ö. Yurttaş, Geometric intersection of curves on punctured disks, Journal of the Mathematical Society of Japan, 65 (2013), 1554-1564. doi: 10.2969/jmsj/06541153.

show all references

References:
[1]

E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, 4 (1925), 47-72. doi: 10.1007/BF02950718.

[2]

E. Artin, Theory of braids, Ann. of Math. (2), 48 (1947), 101-126. doi: 10.2307/1969218.

[3]

M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology, 34 (1995), 109-140. doi: 10.1016/0040-9383(94)E0009-9.

[4]

J. Birman, P. Brinkmann and K. Kawamuro, A polynomial invariant of pseudo-Anosov maps, Journal of Topology and Analysis, 4 (2012), 13-47. doi: 10.1142/S1793525312500033.

[5]

P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest, Why are Braids Orderable?, vol. 14 of Panoramas et Synthèses [Panoramas and Syntheses], Société Mathématique de France, Paris, 2002.

[6]

P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest, Ordering Braids, vol. 148 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/surv/148.

[7]

I. Dynnikov, On a Yang-Baxter mapping and the Dehornoy ordering, Uspekhi Mat. Nauk, 57 (2002), 151-152. doi: 10.1070/RM2002v057n03ABEH000519.

[8]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston Sur les Surfaces, vol. 66 of Astérisque, Société Mathématique de France, Paris, 1979, Séminaire Orsay.

[9]

J. Franks and M. Misiurewicz, Cycles for disk homeomorphisms and thick trees, in Nielsen theory and dynamical systems (South Hadley, MA, 1992), vol. 152 of Contemp. Math., Amer. Math. Soc., Providence, RI, (1993), 69-139. doi: 10.1090/conm/152/01319.

[10]

T. Hall, Software available for download from:, , (). 

[11]

T. Hall and S. Ö. Yurttaş, On the topological entropy of families of braids, Topology Appl., 156 (2009), 1554-1564. doi: 10.1016/j.topol.2009.01.005.

[12]

J. Los, Pseudo-Anosov maps and invariant train tracks in the disc: A finite algorithm, Proc. London Math. Soc. (3), 66 (1993), 400-430. doi: 10.1112/plms/s3-66.2.400.

[13]

L. Mosher, Train track expansions of measured foliations, 2003,, Preprint available from , (). 

[14]

J.-O. Moussafir, On computing the entropy of braids, Funct. Anal. Other Math., 1 (2006), 37-46.

[15]

R. C. Penner and J. L. Harer, Combinatorics of Train Tracks, vol. 125 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1992.

[16]

E. Rykken, Expanding factors for pseudo-anosov homeomorphisms, Rocky Mountain J. Math., 28 (1998), 1103-1124. doi: 10.1216/rmjm/1181071758.

[17]

W. P. Thurston, n the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. doi: 10.1090/S0273-0979-1988-15685-6.

[18]

S. Ö. Yurttaş, Geometric intersection of curves on punctured disks, Journal of the Mathematical Society of Japan, 65 (2013), 1554-1564. doi: 10.2969/jmsj/06541153.

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