Topological entropy of braids on the torus

SIAM Journal on Applied Dynamical Systems

Volumn 6, Issue 1, 2007, Pages 79-98

  Finn, Matthew D ^a   Thiffeault, Jean Luc ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Braid groups; Topological entropy

Indexed keywords

CONTROL THEORY; CONVERGENCE OF NUMERICAL METHODS; ENTROPY; TOPOLOGY;

BRAID GROUPS; EXPONENTIAL CONVERGENCES; FAST METHODS; LOWER BOUNDS; MATRIX METHODS; MATRIX REPRESENTATIONS; PARTICLE TRAJECTORIES; TOPOLOGICAL ENTROPIES; TOPOLOGICAL ENTROPY;

WEAVING;

EID: 54749147921     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/060659636     Document Type: Article

References (32)

1. H. AREF, The development of chaotic advection, Phys. Fluids, 14 (2002), pp. 1315-1325.
2. M. BESTVINA AND M. HANDEL, Train-tracks for surface horneomorphisms, Topology, 34 (1995), pp. 109-140.
3. J. S. BIRMAN, On braid groups, Comm. Pure Appl. Math., 22 (1969), pp. 41-72.
4. J. S. BIRMAN, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud., Princeton University Press, Princeton, NJ, 1975.
5. R. BOWEN, Entropy and the fundamental group, in Structure of Attractors, Lecture Notes in Math. 668, Springer, New York, 1978, pp. 21-29.
6. P. L. BOYLAND, Topological methods in surface dynamics, Topology Appl., 58 (1994), pp. 223-298.
7. P. L. BOYLAND, H. AREF, AND M. A. STREMLER, Topological fluid mechanics of stirring, J. Fluid Mech., 403 (2000), pp. 277-304.
8. D. D'ALESSANDRO, M. DAHLEH, AND I. MEZIĆ , Control of mixing in fluid flow: A maximum entropy approach, IEEE Trans. Automat. Control, 44 (1999), pp. 1852-1863.
9. I. A. DYNNIKOV, On a Yang-Baxter m,ap and the Dehornoy ordering, Russian Math. Surveys, 57 (2002), pp. 592-594.
10. A. FATHI, F. LAUNDENBACH, AND V. POENARU, Travaux de Thurston sur les surfaces, Astérisque, 66-67 (1979), pp. 1-284.
11. M. D. FINN, S. M. COX, AND H. M. BYRNE, Topological chaos in inviscid and viscous mixers, J. Fluid Mech., 493 (2003), pp. 345-361.
12. M. D. FINN, S. M. COX, AND H. M. BYRNE, Mixing measures for a two-dimensional chaotic Stokes flow, J. Engrg. Math., 48 (2004), pp. 129-155.
13. M. D. FINN, J.-L. THIFFEAULT, AND E. GOUILLART, Topological chaos in spatially periodic mixers, Phys. D, 221 (2006), pp. 92-100.
14. J. M. FRANKS AND M. HANDEL, Entropy and exponential growth of π1 in dimension two, Proc. Amer. Math. Soc., 102 (1988), pp. 753-760.
15. J.-M. GAMBAUDO AND E. E. PÉCOU, Dynamical cocycles with values in the Artin braid group, Ergodic Theory Dynam. Systems, 19 (1999), pp. 627-641.
16. E. GOUILLART, J.-L. THIFFEAULT, AND M. D. FINN, Topological mixing with ghost rods, Phys. Rev. E (3), 73 (2006), paper 036311.
17. T. HALL, Train: A C+ + Program for Computing Train Tracks of Surface Hom,eomorphisms, online at http://www.liv.ac.uk/maths/PURE/MIN-SET/ CONTENT/members/T-Hall.html.
18. H. HAMIDI-TEHRANI AND Z.-H. CHEN, Surface diffeomorphisms via train-tracks, Topology Appl., 73 (1996), pp. 141-167.
19. B. KOLEV, Entropie topologique et représentation de Burau, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), pp. 835-838; English translation available at http://arxiv.org/abs/math.DS/0304105.
20. M. LEFRANC, Alternative determinism principle for topological analysis of chaos, Phys. Rev. E, 64 (2006), paper 035202.
21. J.-O. MOUSSAFIR, On the Entropy of Braids, manucript, 2006; available online at http://arxiv.org/abs/math.DS/0603355.
22. S. NEWHOUSE AND T. PIGNATARO, On the estimation of topological entropy, J. Statist. Phys., 72 (1993), pp. 1331-1351.
23. J. NIELSEN, Untersuchungen zur Topologie des geschlossenen zweiseitigen Flächen. I, Acta Math., 50 (1927), pp. 189-358.
24. J. NIELSEN, Untersuchungen zur Topologie des geschlossenen zweiseitigen Flächen. II, Acta Math., 53 (1929), pp. 1-76.
25. J. NIELSEN, Untersuchungen zur Topologie des geschlossenen zweiseitigen Flächen. III, Acta Math., 58 (1932), pp. 87-167.
26. R. C. PENNER, The action of the mapping class group on curves in surfaces, Enseign. Math. (2), 30 (1984), pp. 39-55.
27. R. C. PENNER AND J. L. HARER, Combinatorics of Train Tracks, Ann. of Math. Stud. 125, Princeton University Press, Princeton, NJ, 1991.
28. R. T. PIERREHUMBERT, Tracer microstructure in the large-eddy dominated regime, Chaos Solitons Fractals, 4 (1994), pp. 1091-1110.
29. M. SCHAEFER, D. ŠTEFANKOVIČ, AND E. SEDGWICK, Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time, Technical Report TR 05-009, DePaul University, Chicago, IL, 2005.
30. J.-L. THIFFEAULT, Measuring topological chaos, Phys. Rev. Lett., 94 (2005), paper 084502.
31. J.-L. THIFFEAULT AND M. D. FINN, Topology, braids, and mixing in fluids, Philos. Trans. Roy. Soc. London Ser. A, 364 (2006), pp. 3251-3266.
32. W. THURSTON, On the geometry and dynamics of diffeomorphism,s of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), pp. 417-431.