Hostname: page-component-5cf477f64f-h6p2m Total loading time: 0 Render date: 2025-04-02T04:04:31.287Z Has data issue: false hasContentIssue false
  • English
  • Français

On topological entropy of triangular maps of the square

Published online by Cambridge University Press:  17 April 2009

Lluís Alsedà ,
Sergiĭ F. Kolyada  and
Ľubomír Snoha
Lluís Alsedà
Affiliation:
Departament de Matemàtiques, Universitat Autònoma, de Barcelona 08193 – Bellaterra Barcelona, Spain
Sergiĭ F. Kolyada
Affiliation:
Department of Mathematics Faculty of Education Tajovského 40 975 49 Banská Bystrica, Czechoslovakia
Ľubomír Snoha
Affiliation:
Institute of Mathematics, Ukrainian Academy of Sciences Repin str.3 252601 Kiev - 4, Ukraine
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the topological entropy of triangular maps of the square. We show that such maps differ from the continuous maps of the interval because there exist triangular maps of the square of “type 2∞” with infinite topological entropy. The set of such maps is dense in the space of triangular maps of “type at most 2∞” and the topological entropy as a function of the triangular maps of the square is not lower semicontinuous. However, we show that for these maps the characterisation of the lower bounds of the topological entropy depending on the set of periods is the same as for the continuous maps of the interval.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 1 , August 1993 , pp. 55 - 67
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Adler, R., Konheim, A. and McAndrew, J., ‘Topological entropy’, Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
[2]Alsedà, L., Llibre, J. and Misiurewicz, M., Combinatorial dynamics and entropy in dimension one (World Scientific, Singapore, 1993).CrossRefGoogle Scholar
[3]Alsedà, L. and Moreno, J.M., ‘Lower bounds of topological entropy of maps of Y’, J. Math. Anal. Appl. 155 (1991), 512–530.CrossRefGoogle Scholar
[4]Block, L. and Coppel, W.A., Dynamics in one dimension, Lecture Notes in Math. 1513 (Springer-Verlag, Berlin, Heidelberg, New York, 1992).CrossRefGoogle Scholar
[5]Block, L., Guckenheimer, J., Misiurewicz, M. and Young, L.S., ‘Periodic points and topological entropy of one dimensional maps’, in Global theory of dynamical systems, Lecture Notes in Math. 819 (Springer-Verlag, Berlin, Hedelberg, New York, 1980), pp. 18–34.CrossRefGoogle Scholar
[6]Blokh, A.M., ‘On some properties of graph maps; spectral decomposition, Misiurewicz conjecture and abstract sets of periods’, Preprint #35, Max-Plank-Institut für Mathematik.Google Scholar
[7]Bowen, R. and Franks, J., ‘The periodic points of maps of the disk and the interval’, Topology 15 (1976), 337–342.CrossRefGoogle Scholar
[8]Bowen, R., ‘Entropy for group endomorphisms and homogeneous spaces’, Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
[9]Bowen, R., ‘Periodic points and measures for axiom A diffeomorphisms’, Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
[10]Dinaburg, E.I., ‘Connection between various entropy characterizations of dynamical systems’, (Russian), Izv. Akad. Nauk. SSSR 35 (1971), 324–366.Google Scholar
[11]Gambaudo, J.M., van Strien, S. and Tresser, C., ‘The periodic orbit structure of orientation preserving diffeomorphisms on D 2 with topological entropy zero’, Ann. Inst. H. Poincaré Phys. Théor. 49 (1989), 335–356.Google Scholar
[12]Jonker, L. and Rand, D., ‘The periodic orbits and entropy of certain maps of the unit interval’, J. London Math. Soc. 22 (1980), 175–181.CrossRefGoogle Scholar
[13]Jonker, L. and Rand, D., ‘Une borne inferieure pour l’entropie de certaines applications de l’intervalle dans lui-même’, C.R. Acad. Set., Paris, Sér. I Math. 287 (1978), 501–502.Google Scholar
[14]Kloeden, P.E., ‘On Sharkovsky’s cycles coexistence ordering’, Bull. Austral. Math. Soc. 20 (1979), 171–177.CrossRefGoogle Scholar
[15]Kolyada, S.F., ‘On triangular maps of type 2∞ with positive entropy’, (Russian), in Dynamical systems and turbulence, Inst. Math. Ukrain. Acad. Sci. Kiev., 1989, pp. 76–82.Google Scholar
[16]Kolyada, S.F., ‘On dynamics of triangular maps of the square’, Preprint 91.14. Inst. Math. Ukrain. Acad. Sci., Kiev, 1991, Ergodic Theory and Dynamical Systems (to appear).Google Scholar
[17]Kolyada, S.F. and Sharkovsky, A.N., ‘On topological dynamics of triangular maps of the plane’, in Proc. ECIT-89 (World Scientific, Singapore, 1991), pp. 177–183.Google Scholar
[18]Li, T.Y., Misiurewicz, M., Pianigiani, G. and Yorke, J.A., ‘No division implies chaos’, Trans. Amer. Math. Soc. 273 (1982), 191–199.CrossRefGoogle Scholar
[19]Llibre, J. and Mackay, R.S., ‘A classification of braid types for diffeomorphisms of surfaces of genus zero with topological entropy zero’, J. London Math. Soc. (2) 42 (1990), 562–576.CrossRefGoogle Scholar
[20]Libre, J. and Misiurewicz, M., ‘Horseshoes, entropy and periods for graph maps’, Topology (to appear).Google Scholar
[21]Misiurewicz, M., ‘Horseshoes for mappings of an interval’, Bull. Acad. Pol. Sci., Ser. Sci. Math. 27 (1979), 167–169.Google Scholar
[22]Sharkovsky, A.N., ‘Coexistence of cycles of a continuous map of the line into itself’, (Russian), Ukrain. Math. Zh. 16 (1964), 61–71.Google Scholar
[23]Sharkovsky, A.N., Kolyada, S.F., Sivak, A.G. and Fedorenko, V.V., ‘Dynamics of one- dimensional maps’, (Russian), Naukova Dumka. Kiev (1989).Google Scholar