Hostname: page-component-76dd75c94c-qmf6w Total loading time: 0 Render date: 2024-04-30T09:33:27.176Z Has data issue: false hasContentIssue false
  • English
  • Français

Embedding the symbolic dynamics of Lorenz maps

Published online by Cambridge University Press:  03 March 2014

TONY SAMUEL ,
NINA SNIGIREVA  and
ANDREW VINCE
TONY SAMUEL
Affiliation:
Fachbereich 3 – Mathematik, Universität Bremen, 28359 Bremen, Germany. e-mail: tony@math.uni-bremen.de
NINA SNIGIREVA
Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland. e-mail: nina.snigireva@ucd.ie
ANDREW VINCE
Affiliation:
Department of Mathematics, University of Florida, Gaineville, Florida, U.S.A. e-mail: avince@ufl.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Necessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 3 , May 2014 , pp. 505 - 519
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alsedà, L. and Mañosas, F.Kneading theory for a family of circle maps with one discontinuity. Acta Math. Univ. Comenian. (N.S.) 65 (1996), 11–22.Google Scholar
[2]Barnsley, M. F., Harding, B. and Igudesman, K.How to transform and filter images using iterated function systems. Society for Industrial and Applied Mathematics 4 (2011), 1001–1028.Google Scholar
[3]Barnsley, M. F., Harding, B. and Vince, A. The entropy of an overlapping dynamical system. Ergodic Theory Dynam. Systems, to appear (2011).Google Scholar
[4]Block, L. and Keesling, J.Computing the topological entropy of maps of the interval with three monotone pieces. J. Stat. Phys. 66 (1992), 755–774.Google Scholar
[5]Block, L., Keesling, J., Li, S. and Peterson, K.An improved algorithm for computing topological entropy. J. Stat. Phys. 55 (1989), 929–939.Google Scholar
[6]Brin, M. and Stuck, G.Introduction to Dynamical Systems (Cambridge University Press, 2002).Google Scholar
[7]Dajani, K. and Kraaikamp, C.Ergodic Theory of Numbers. Carus Math. Monogr., no. 29 (2002).Google Scholar
[8]Eckhardt, B. and Ott, G.Periodic orbit analysis of the Lorenz attractor. Zeit. Phys. B 93 (1994), 258–266.Google Scholar
[9]Froyland, G., Murray, R. and Terhesiu, D.Efficient computation of topological entropy, pressure, conformal measures and equilibrium states in one dimension. Phys. Rev. E 76 (2007).CrossRefGoogle ScholarPubMed
[10]Glendinning, P.Topological conjugation of Lorenz maps by β-transformations. Math. Proc. Camb. Phil. Soc. 107 (1990), 401–413.CrossRefGoogle Scholar
[11]Glendinning, P. and Hall, T.Zeros of the kneading invariant and topological entropy for Lorenz maps. Nonlinearity. 9 (1996), 999–1014.Google Scholar
[12]Glendinning, P. and Sparrow, C.Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps. Phys. D. 62 (1993), 22–50.Google Scholar
[13]Hofbauer, F.Maximal measures for piecewise monotonically increasing transformations on [0,1]. Lecture Notes in Math. vol. 729 (Springer-Verlag, 1979), 66–77.Google Scholar
[14]Hofbauer, F. and Raith, P.The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval. Canad. Math. Bull. 35 (1992), 84–98.Google Scholar
[15]Hubbard, J. H. and Sparrow, C.The classification of topologically expansive Lorenz maps. Comm. Pure Appl. Math. 43 (1990), 431–443.Google Scholar
[16]Lorenz, E. N.Deterministic nonperiodic flow. J. Atmos. Sci. 20 (1963), 130–141.Google Scholar
[17]Miilnor, J. Is entropy effectively computable. www.math.iupui.edu/~mmisiure/open (2002).Google Scholar
[18]Milnor, J. and Thurston, W.On iterated maps of the interval. Lecture Notes in Math. vol. 67 (Springer-Verlag, 1980), 45–63.Google Scholar
[19]Misiurewicz, M.Possible jumps of entropy for interval maps. Qual. Theory Dyn. Syst. 2 (2001), 289–306.Google Scholar
[20]Parry, W.Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1965), 368–378.Google Scholar
[21]Viswanath, D.Symbolic dynamics and periodic orbits of the Lorenz attractor. Nonlinearity 16 (2003), 1035–1056.CrossRefGoogle Scholar
[22]Williams, R. F.Structure of Lorenz attractors. Publ. Math. Inst. Hautes Études Sci. 50 (1980), 73–100.Google Scholar