Hostname: page-component-76d6cb85b7-5qg8f Total loading time: 0 Render date: 2026-07-16T04:57:44.668Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical itineraries of maps with constant slope and one discontinuity

Published online by Cambridge University Press:  24 November 2014

MICHAEL BARNSLEY ,
WOLFGANG STEINER  and
ANDREW VINCE
MICHAEL BARNSLEY
Affiliation:
The Australian National University, Canberra, Australia. e-mail: michael.barnsley@anu.edu.au
WOLFGANG STEINER
Affiliation:
LIAFA, CNRS UMR 7089, Université Paris Diderot – Paris 7, Paris, France. e-mail: steiner@liafa.univ-paris-diderot.fr
ANDREW VINCE
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL, U.S.A. e-mail: avince@ufl.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

For a function from the unit interval to itself with constant slope and one discontinuity, the itineraries of the point of discontinuity are called the critical itineraries. These critical itineraries play a significant role in the study of β-expansions (with positive or negative β) and fractal transformations. A combinatorial characterization of the critical itineraries of such functions is provided.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 3 , November 2014 , pp. 547 - 565
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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Barnsley, M. F.Transformations between self-referential sets. Amer. Math. Monthly 116 (2009), no. 4, 291–304.CrossRefGoogle Scholar
[2]Barnsley, M. F., Harding, B. and Igudesman, K.How to transform and filter images using iterated function systems. SIAM J. Imaging 4 (2011), no. 4, 1001–1028.Google Scholar
[3]Barnsley, M. F., Harding, B. and Vince, A.The entropy of a special overlapping dynamical system. Ergodic Theory Dynam. Systems 34 (2014), no. 2, 469–486.Google Scholar
[4]Collet, P. and Eckmann, J. P.Iterated Maps on the Interval as Dynamical Systems (Birkhäuser, Boston, 1980).Google Scholar
[5]Dajani, K. and Kalle, C.Transformations generating negative β-expansions. Integers 11B (2011), Paper No. A5, 18.Google Scholar
[6]Flatto, L. and Lagarias, J. C.The lap-counting function for linear mod one transformations. I. Explicit formulas and renormalizability. Ergodic Theory Dynam. Systems 16 (1996), no. 3, 451–491.Google Scholar
[7]Glendinning, P.Topological conjugation of Lorenz maps by β-transformations. Math. Proc. Camb. Phil. Soc. 107 (1990), no. 2, 401–413.CrossRefGoogle Scholar
[8]Glendinning, P. and Hall, T.Zeros of the kneading invariant and topological entropy for Lorenz maps. Nonlinearity 9 (1996), no. 4, 999–1014.Google Scholar
[9]Glendinning, P. and Sparrow, C.Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps. Phys. D 62 (1993), no. 1–4, 22–50.Google Scholar
[10]Hofbauer, F.The maximal measure for linear mod one transformations. J. London Math. Soc. (2) 23 (1981), no. 1, 92–112.Google Scholar
[11]Hogan, S. J., Higham, L. and Griffin, T. C. L.Dynamics of a piecewise linear map with a gap. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2077, 49–65.Google Scholar
[12]Hubbard, J. H. and Sparrow, C. T.The classification of topologically expansive Lorenz maps. Comm. Pure Appl. Math. 43 (1990), no. 4, 431–443.Google Scholar
[13]Ito, S. and Sadahiro, T.Beta-expansions with negative bases. Integers 9 (2009), A22, 239–259.Google Scholar
[14]Jain, P. and Banerjee, S.Border-collision bifurcations in one-dimensional discontinuous maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 11, 3341–3351.Google Scholar
[15]Kalle, C. and Steiner, W.Beta-expansions, natural extensions and multiple tilings associated with Pisot units. Trans. Amer. Math. Soc. 364 (2012), no. 5, 2281–2318.CrossRefGoogle Scholar
[16]Keener, J. P.Chaotic behavior in piecewise continuous difference equations. Trans. Amer. Math. Soc. 261 (1980), no. 2, 589–604.Google Scholar
[17]Liao, L. and Steiner, W.Dynamical properties of the negative beta-transformation. Ergodic Theory Dynam. Systems 32 (2012), no. 5, 1673–1690.CrossRefGoogle Scholar
[18]Lind, D. and Marcus, B.An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995).Google Scholar
[19]Lorenz, E. N.Deterministic nonperiodic flows. Atmospheric Sciences 20 (1963), no. 2, 130–141.Google Scholar
[20]Milnor, J. and Thurston, W.On iterated maps of the interval. Dynamical systems (College Park, MD, 1986–87), Lecture Notes in Math. vol. 1342 (Springer, Berlin, 1988), pp. 465–563.Google Scholar
[21]Parry, W.On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.Google Scholar
[22]Parry, W.Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 368–378.Google Scholar
[23]Rényi, A.Representations for real numbers and thier ergodic properties. Acta Math. Acad. Sci.Hungar. 8 (1957), 477–493.CrossRefGoogle Scholar
[24]Sharkovsky, A. N. and Chua, L. O.Chaos in some 1-d discontinuous maps that appear in the analysis of electrical circuits. IEEE Trans. Circuits Systems I Fund. Theory Appl. 40 (1993), no. 10, 722–731.Google Scholar
[25]Shultz, F.Dimension groups for interval maps. II. The transitive case. Ergodic Theory Dynam. Systems 27 (2007), no. 4, 1287–1321.Google Scholar
[26]Silva, L. and Sousa Ramos, J.Topological invariants and renormalization of Lorenz maps. Phys. D 162 (2002), no. 3-4, 233–243.CrossRefGoogle Scholar
[27]Steiner, W.Digital expansions with negative real bases. Acta Math. Hungar. 139 (2013), no. 1–2, 106–119.Google Scholar