Topological Methods in Nonlinear Analysis

Isolated sets, catenary Lyapunov functions and expansive systems

Alfonso Artigue

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


It is a paper about models for isolated sets and the construction of special hyperbolic Lyapunov functions. We prove that after a suitable surgery every isolated set is the intersection of an attractor and a repeller. We give linear models for attractors and repellers. With these tools we construct hyperbolic Lyapunov functions and metrics around an isolated set whose values along the orbits are catenary curves. Applications are given to expansive flows and homeomorphisms, obtaining, among other things, a hyperbolic metric on local cross sections for an arbitrary expansive flow on a compact metric space.

Article information

Topol. Methods Nonlinear Anal., Volume 49, Number 1 (2017), 21-50.

First available in Project Euclid: 11 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Artigue, Alfonso. Isolated sets, catenary Lyapunov functions and expansive systems. Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 21--50. doi:10.12775/TMNA.2016.067.

Export citation


  • F. Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (2003), 14–67.
  • A. Artigue, Finite sets with fake observable cardinality, Bull. Korean Math. Soc. 52 (2015), 323–333.
  • J. Auslander and P. Seibert, Prolongations and stability in dynamical systems, Ann. Inst. Fourier 14 (1964), 237–267.
  • N.P. Bhatia and G.P. Szegö, Dynamical Systems: Stability Theory and Applications, vol. 35, Springer–Verlag, Lecture Notes in Math., 1967.
  • R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc. 164 (1972), 323–331.
  • R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations 12 (1972), 180–193.
  • C. Conley, Isolated invariant sets and the Morse index, Conference Board of the Mathematical Sciences, 38 (1978), Amer. Math. Soc.
  • ––––, The gradient structure of a flow \romI, Ergodic Theory Dynam. Systems 8 (1988), 11–26.
  • C. Conley and R. Easton, Isolated Invariant Sets and Isolated Blocks, Trans. Amer. Math. Soc. 158 (1971).
  • R.C. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330–352.
  • J. Denzler and A.M. Hinz, Catenaria vera – the true catenary, Expo. Math. 17 (1999), 117–142.
  • A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys. 126 (1989), 249–262.
  • K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), 117–162.
  • J. Hocking and G. Young, Topology, Addison–Wesley Publishing Company, Inc., 1961.
  • M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. of the Am. Math. Soc. 126 (1998), 245–256.
  • A. Illanes and S.B. Nadler Jr., Hyperspaces: Fundamentals and Recent Advances, Marcel Dekker, Inc., 1999.
  • H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576–598.
  • N.N. Krasovskiĭ, Stability of Motion, Stanford University Press, 1963.
  • J. Lewowicz, Lyapunov functions and topological stability, J. Differential Equations 38 (1980), 192–209.
  • ––––, Expansive homeomorphisms of surfaces, Bull. Braz. Math. Soc. 20 (1989), 113–133.
  • J.L. Massera, On Liapunoff's conditions of stability, Ann. of Math. 50 (1949), 705–721.
  • C.A. Morales, A generalization of expansivity, Discrete Contin. Dynam. Systems 32 (2012), 293–301.
  • M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc. 24 (1993), 179–199.
  • W.L. Reddy, Pointwise expansion homeomorphisms, J. London Math. Soc. 2 (1970), 232–236.
  • R. Ures, On expansive covering maps, Publ. Mat. Urug. 3 (1990), 59–67.
  • W.R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769–774.
  • J.L. Vieitez, Lyapunov functions and expansive diffeomorphisms on \rom3D-manifolds, Ergodic Theory Dynam. Systems 22 (2002), 601–632.
  • H. Whitney, Regular families of curves, Ann. of Math. 34 (1933), 244–270.
  • F.W. Wilson Jr. and J.A. Yorke, Lyapunov functions and isolating blocks, J. Differential Equations 13 (1973), 106–123.