Rad HAZU, Matematičke znanosti, Vol. 21 (2017), 117-141.

A NEW MEASURE OF INSTABILITY AND TOPOLOGICAL ENTROPY OF AREA-PRESERVING TWIST DIFFEOMORPHISMS

Siniša Slijepčević

Department of Mathematics, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: slijepce@math.hr

Abstract.   We introduce a new measure of instability of area-preserving twist diffeomorphisms, which generalizes the notions of angle of splitting of separatrices, and flux through a gap of a Cantori. As an example of application, we establish a sharp > 0 lower bound on the topological entropy in a neighbourhood of a hyperbolic, unique action-minimizing fixed point, assuming only no topological obstruction to diffusion, i.e. no homotopically non-trivial invariant circle consisting of orbits with the rotation number 0. The proof is based on a new method of precise construction of positive entropy invariant measures, applicable to more general Lagrangian systems, also in higher degrees of freedom.

2010 Mathematics Subject Classification.   37E40 (primary); 37J45, 37A35 (secondary).

Key words and phrases.   Twist maps, topological entropy, metric entropy, separatrix splitting, variational techniques.

Full text (PDF) (free access)

DOI: https://doi.org/10.21857/m8vqrt0z59

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