Isolated sets, catenary Lyapunov functions and expansive systems

Abstract

Abstract. 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.

Recent developments around partial actions

“…The importance of the globalization problem lies in the possibility to relate partial actions with global ones and this way try to move from global results to the partial setting, producing more general facts, as well as to obtain applications to the global case in situations in which partial actions appear naturally, as it occurred in [23]. Thus facts about globalization from [4] were used in [237] with respect to K -theory of reduced C * -algebras of 0-F-inverse semigroups, in [225] in the K -theoretic study of reduced crossed products attached to totally disconnected dynamical systems, and in [26] for partial flows with application to Lyapunov functions. In addition, globalizable partial actions were essential for the development of Galois Theory of partial group actions in [123], for the elaboration of the concept of a partial Hopf (co)action in [72], as well as in a series of ring theoretic and Galois theoretic investigations in [27,29,30,32,35,39,41,49,50,69,77,98,99,101,104,106,171,176,252,253].…”

Recent developments around partial actions

“…The importance of the globalization problem lies in the possibility to relate partial actions with global ones and this way try to move from global results to the partial setting, producing more general facts, as well as to obtain applications to the global case in situations in which partial actions appear naturally, as it occurred in [23]. Thus facts about globalization from [4] were used in [237] with respect to K-theory of reduced C * -algebras of 0-F -inverse semigroups, in [225] in the K-theoretic study of reduced crossed products attached to totally disconnected dynamical systems, and in [26] for partial flows with application to Lyapunov functions. In addition, globalizable partial actions were essential for the development of Galois Theory of partial group actions in [123], for the elaboration of the concept of a partial Hopf (co)action in [72], as well as in a series of ring theoretic and Galois theoretic investigations in [27], [29], [30], [32], [35], [39], [41], [49], [50], [69], [77], [98], [99], [101], [104], [106], [171], [176], [252], [253].…”