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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/BF02950718. Google Scholar

[2]

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

[3]

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

[4]

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

[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, (2002). Google Scholar

[6]

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

[7]

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

[8]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston Sur les Surfaces, vol. 66 of Astérisque,, Société Mathématique de France, (1979). Google Scholar

[9]

J. Franks and M. Misiurewicz, Cycles for disk homeomorphisms and thick trees,, in Nielsen theory and dynamical systems (South Hadley, (1993), 69. doi: 10.1090/conm/152/01319. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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, (2002). Google Scholar

[6]

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

[7]

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

[8]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston Sur les Surfaces, vol. 66 of Astérisque,, Société Mathématique de France, (1979). Google Scholar

[9]

J. Franks and M. Misiurewicz, Cycles for disk homeomorphisms and thick trees,, in Nielsen theory and dynamical systems (South Hadley, (1993), 69. doi: 10.1090/conm/152/01319. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

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