# A new measure of instability and topological entropy of area-preserving twist diffeomorphisms

## TL;DR

This paper introduces a new measure of instability for area-preserving twist diffeomorphisms, providing a lower bound on topological entropy near a hyperbolic fixed point and developing a novel method for constructing positive entropy measures.

## Contribution

It presents a novel instability measure that generalizes existing concepts and establishes a sharp lower bound on topological entropy under minimal assumptions.

## Key findings

- Established a positive lower bound on topological entropy near a hyperbolic fixed point.
- Developed a new method for constructing positive entropy invariant measures.
- Applicable to general Lagrangian systems with higher degrees of freedom.

## 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.