Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T00:34:04.722Z Has data issue: false hasContentIssue false
  • English
  • Français

Dendritations of surfaces

Published online by Cambridge University Press:  03 April 2017

ALFONSO ARTIGUE Open the ORCID record for ALFONSO ARTIGUE [Opens in a new window]
ALFONSO ARTIGUE*
Affiliation:
Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto-Uruguay, Uruguay email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we develop a generalization of foliated manifolds in the context of metric spaces. In particular we study dendritations of surfaces that are defined as maximal atlases of compatible upper semicontinuous local decompositions into dendrites. Applications are given in modeling stable and unstable sets of topological dynamical systems. For this purpose new forms of expansivity are defined.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 8 , December 2018 , pp. 2860 - 2912
Copyright
© Cambridge University Press, 2017 

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

Acosta, G., Hernández-Gutiérrez, R., Naghmouchi, I. and Oprocha, P.. Periodic points and transitivity on dendrites. Ergod. Th. & Dynam. Sys., to appear. doi:10.1017/etds.2015.137, published online 8 March 2016.Google Scholar
Aoki, N. and Hiraide, K.. Topological Theory of Dynamical Systems—Recent Advances (North-Holland Mathematical Library, 52) . North-Holland, Amsterdam, 1994.Google Scholar
Artigue, A.. Kinematic expansive flows. Ergod. Th. & Dynam. Sys. 36 (2016a), 390–421.Google Scholar
Artigue, A.. Robustly N-expansive surface diffeomorphisms. Discrete Contin. Dyn. Syst. 36 (2016b), 2367–2376.Google Scholar
Artigue, A.. Anomalous cw-expansive surface homeomorphisms. Discrete Contin. Dyn. Syst. 36 (2016c), 3511–3518.Google Scholar
Artigue, A., Pacífico, M. J. and Vieitez, J. L.. N-expansive homeomorphisms on surfaces. Commun. Contemp. Math. 19 (2017), 1–18.Google Scholar
Barge, M. and Martensen, B. F.. Classification of expansive attractors on surfaces. Ergod. Th. & Dynam. Sys. 31 (2011), 1619–1639.Google Scholar
Bowen, R. and Walters, P.. Expansive one-parameter flows. J. Differential Equations 12 (1972), 180–193.Google Scholar
Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2003.Google Scholar
Camacho, C. and Neto, A. L.. Geometric Theory of Foliations. Birkhäuser, Boston, MA, 1985.Google Scholar
Candel, A. and Conlon, L.. Foliations I (Graduate Studies in Mathematics, 23) . American Mathematical Society, Providence, RI, 2000.Google Scholar
de Carvalho, A. and Paternain, M.. Monotone quotients of surface diffeomorphisms. Math. Res. Lett. 10 (2003), 603–619.Google Scholar
Charatonik, J. J. and Charatonik, W. J.. Connected subsets of dendrites and separators of the plane. Topology Appl. 36 (1990), 233–245.Google Scholar
Daverman, R. J.. Decompositions of Manifolds (Pure and Applied Mathematics, 124) . Academic Press, Cambridge, MA, 1986.Google Scholar
Engelking, R.. General Topology (Sigma Series in Pure Mathematics, 6) . Heldermann, Berlin, 1989.Google Scholar
Franks, J. and Robinson, C.. A quasi-Anosov diffeomorphism that is not Anosov. Trans. Amer. Math. Soc. 223 (1976), 267–278.Google Scholar
Hiraide, K.. Expansive homeomorphisms with the pseudo-orbit tracing property of n-tori. J. Math. Soc. Japan 41 (1989), 357–389.Google Scholar
Hiraide, K.. Expansive homeomorphisms of compact surfaces are pseudo-Anosov. Osaka J. Math. 27 (1990), 117–162.Google Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.Google Scholar
Hocking, J. G. and Young, G. S.. Topology. Addison-Wesley, London, 1961.Google Scholar
Hurewicz, W. and Wallman, H.. Dimension Theory, Revised edn. Princeton University Press, Princeton, NJ, 1948.Google Scholar
Jones, F. B.. On the existence of a small connected open set with a connected boundary. Bull. Amer. Math. Soc. (N.S.) 68 (1962), 117–119.Google Scholar
Kato, H.. Continuum-wise expansive homeomorphisms. Canad. J. Math. 45 (1993a), 576–598.Google Scholar
Kato, H.. Concerning continuum-wise fully expansive homeomorphisms of continua. Topology Appl. 53 (1993b), 239–258.Google Scholar
Kawamura, K., Tuncali, H. M. and Tymchatyn, E. D.. Expansive homeomorphisms on Peano curves. Houston J. Math. 21 (1995), 573–583.Google Scholar
Komuro, M.. Expansive properties of Lorenz attractors. The Theory of Dynamical Systems and its Applications to Nonlinear Problems. World Scientific, Singapore, 1984, pp. 4–26.Google Scholar
Kuratowski, K.. Introduction to Set Theory and Topology. Pergamon Press, Oxford, 1961.Google Scholar
Kuratowski, K.. Topology Vol. II. Academic Press, New York, 1968.Google Scholar
Lee, J. M.. Introduction to Topological Manifolds (Graduate Texts in Mathematics, 202) . Springer, New York, 2000.Google Scholar
Lewowicz, J.. Expansive homeomorphisms of surfaces. Bol. Soc. Bras. Mat. 20 (1989), 113–133.Google Scholar
Mañé, R.. Expansive diffeomorphisms. Proc. Symp. on Dynamical Systems (Warwick, UK, 1973/74) (Lecture Notes in Mathematics, 468) . Ed. Anthony, M.. Springer, Berlin, 1975, pp. 162–174.Google Scholar
Mañé, R.. Expansive homeomorphisms and topological dimension. Trans. Amer. Math. Soc. 252 (1979), 313–319.Google Scholar
McMullen, C. T.. Complex Dynamics and Renormalization (Annals of Mathematical Studies, 135) . Princeton University Press, Princeton, NJ, 1994.Google Scholar
Moore, R. L.. Concerning upper-semicontinuous collections of continua. Trans. Amer. Math. Soc. 4 (1925), 416–428.Google Scholar
Moore, R. L.. Concerning triods in the plane and the junction points of plane continua. Proc. Natl Acad. Sci. USA 14 (1928), 85–88.Google Scholar
Morales, C. A.. A generalization of expansivity. Discrete Contin. Dyn. Syst. 32 (2012), 293–301.Google Scholar
Nadler, S. Jr. Hyperspaces of Sets. Marcel Dekker, New York, 1978.Google Scholar
Nadler, S. Jr. Continuum Theory (Pure and Applied Mathematics, 158) . Marcel Dekker, New York, 1992.Google Scholar
Pacifico, M. J., Pujals, E. R. and Vieitez, J. L.. Robustly expansive homoclinic classes. Ergod. Th. & Dynam. Sys. 25 (2005), 271–300.Google Scholar
Pacifico, M. J. and Vieitez, J. L.. Entropy expansiveness and domination for surface diffeomorphisms. Rev. Mat. Complut. 21(2) (2008), 293–317.Google Scholar
Passeggi, A. and Xavier, J.. A classification of minimal sets for surface homeomorphisms. Math. Z. 278 (2014), 1153–1177.Google Scholar
Pittman, C. R.. An elementary proof of the triod theorem. Proc. Amer. Math. Soc. 25 (1970), 919.Google Scholar
Roberts, J. H.. There does not exist an upper semi-continuous decomposition of E into arcs. Duke Math. J. 2 (1936), 10–19.Google Scholar
Sambarino, M.. Estructura local de conjuntos estables e inestables de homeomorfismos en superficies. Universidad de la República, Uruguay, Monograph, 1993.Google Scholar
Smith, M.. A theorem on continuous decompositions of the plane into nonseparating continua. Proc. Amer. Math. Soc. 55 (1976), 221–222.Google Scholar
Vieitez, J. L.. Lyapunov functions and expansive diffeomorphisms on 3D-manifolds. Ergod. Th. & Dynam. Sys. 22 (2002), 601–632.Google Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.Google Scholar
Whitney, H.. Regular families of curves. Ann. of Math. (2) 34 (1933), 244–270.Google Scholar