Topological Entropy of Pseudo-Anosov Braids and Attracting Matrices

Abstract

In this paper, we introduce an alternative method to calculate the topological entropy of each member of an infinite family of pseudo—Anosov braids on the finitely punctured disk making use of 𝛑𝟏- train tracks The method is based on Thurston’s theory of surface homeomorphisms and presents positive matrices alternative to Dynnikov matrices which compute the topological entropy of a given pseudo-Anosov braid.

Keywords

• Topological entropy
• π1- train tracks
• pseudo-Anosov braids

Pseudo-Anosov Örgülerin Topolojik Entropisi ve Çekici Matrisler

Abstract

Bu makalede, sonlu noktası çıkarılmış disk yüzeyinde tanımlı pseudo-Anosov tipinden örgülerin sonsuz bir ailesinin her bir üyesinin topolojik entropisi 𝛑𝟏- train track grafikleri yardımıyla hesaplanmıştır. Kullanılan yöntem Thurston’ın yüzey homeomorfizmaları kuramına dayanmakta ve ilgili pseudo-Anosov örgünün topolojik entropisini veren Dynnikov matrislere alternatif pozitif matrisler sunmaktadır.

Keywords

• Topolojik entropi
• π1- train track grafikleri
• pseudo-Anosov örgüler

References

1. Artin E, 1925 Theorie der Z ̈opfe. Abh. Math. Sem. Univ. Hamburg (4): 47-72.
2. Artin E, 1947 Theory of braids. Ann. of Math. (2), 48:101-126.
3. Bestvina M and Handel M, 1995. Train-tracks for surface homeomorphisms. Topology, 34(1):109-140.
4. Dynnikov I and Wiest B, 2007. On the complexity of braids. J. Eur. Math. Soc. (JEMS), 9(4):801-840.
5. Dynnikov I, 2002. On a Yang-Baxter mapping and the Dehornoy ordering. Us- pekhi Mat. Nauk, 57(3(345)):151-152.
6. Fathi A, Laudenbach F and Poenaru V, 1979. Travaux de Thurston sur les surfaces, volume 66 of Ast ́erisque. Soci ́et ́e Math ́ematique de France, Paris, S ́eminaire Orsay.
7. Hall T and Yurttaş S.Ö, 2009. On the topological entropy of families of braids. Topology Appl., 156(8):1554-1564.
8. Hall T. Software available for download from http://www.maths.liv.ac.uk/~tobyhall/software/.

9. Penner R. C. and Harer J. L, 1992. Combinatorics of train tracks. Annals of Mathematics Studies, volume 125. Princeton University Press, Princeton, NJ.
10. Thurston WP, 1988. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.), 19(2):417-431.
11. Yurttaş SÖ, 2016. Dynnikov and train track transition matrices of pseudo-Anosov braids, Discrete and Continuous Dynamical Systems. Series A, 36(1):109-140.