Real Analysis Exchange

Dynamical systems generated by functions with connected Gδ graphs.

Michaela Čiklová

Full-text: Open access


In 2001, Cs\" ornyei, O'Neil and Preiss proved that the composition of any two Darboux Baire-1 functions $[0,1]\rightarrow [0,1]$ possesses a fixed point, solving a long-standing open problem. In 2004 Szuca proved that this result can be generalized to any $f$ in the class $\cal J$ of functions $[0,1]\rightarrow [0,1]$ with connected $G_\delta$ graph. As a consequence, he proved that for such functions the Sharkovsky theorem is satisfied. As the main result of this paper we prove that as for continuous maps of the interval, any $f$ in $\cal J$ has positive topological entropy if and only if it has a periodic point of period different from $2^n$, for any $n\in\mathbb N$. To do this we show that using Bowen's approach it is possible to define topological entropy for discontinuous maps of a compact metric space with almost all of the standard properties. In particular, the variational principle is true, and consequently, topological entropy is supported by the set of recurrent points. We also develop theory of recurrent, $\omega$-limit, and nonwandering points of functions in $\cal J$ since, in general, standard results from the topological dynamics, are not true. For example, there is a Darboux Baire-1 function $f$ (hence, $f\in\cal J$) such that neither the set of recurrent points nor the set of $\omega$-limit points of $f$ are invariant.

Article information

Real Anal. Exchange, Volume 30, Number 2 (2004), 617 - 638 .

First available in Project Euclid: 15 October 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 37B40: Topological entropy 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25] 54H20: Topological dynamics [See also 28Dxx, 37Bxx]
Secondary: 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05] 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

topological entropy discontinuous maps compact metric space functions with "connected" graph $G_\delta$ sets


Čiklová, Michaela. Dynamical systems generated by functions with connected G δ graphs. Real Anal. Exchange 30 (2004), no. 2, 617 -- 638.

Export citation


  • L. Alsedà, S. Kolyada, J. Llibre and L'. Snoha, Axiomatic definition of the topological entropy on the interval, Aequationes Math., 65 (2003), 113–132.
  • L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lectures Notes in Math. 1513, Springer, Berlin, 1992.
  • A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real analysis, Prentice Hall, Upper Side River, NJ, 1997.
  • M. Csörnyei, T. C. O'Neil and D. Preiss, The composition of two derivatives has a fixed point. Real Anal. Exchange, 26 (2000/01), 749–760.
  • P. S. Keller, Chaotic behavior of Newton's method, Real Anal. Exchange, 18 (1992/93), 490–507.
  • J. Smítal, Chaotic maps with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269–282.
  • K. Smítalová and Š. Šujan, A Mathematical Treatment of Dynamical Models in Biological Science, Ellis Horwood, London, 1991.
  • W. Szlenk, Wstęp do teorii gładkich układów dynamicznych, Pa\' nstwowe Wydawnictwo Naukowe, Warszawa, 1982.
  • P. Szuca, Sharkovskiĭ's theorem holds for some discontinuous functions. Fundam. Math., 179 (2003), 27–41.
  • P. Szuca, Loops of intervals and Darboux Baire-1 fixed point problem. Real Anal. Exchange, 29 (2003/04), 205–210.