Nonuniformly expanding 1d maps with logarithmic singularities

TL;DR

This paper constructs a set of parameters for a family of circle maps with singularities, demonstrating nonuniform expansion and chaotic dynamics, which has implications for understanding complex behaviors in perturbed differential systems.

Contribution

It introduces a method to identify parameters leading to nonuniform expansion in maps with logarithmic singularities, revealing new chaotic dynamics insights.

Findings

Positive measure set of parameters with nonuniform expansion

Existence of chaotic dynamics in perturbed differential equations

Maps exhibit critical points and logarithmic singularities

Abstract

For a certain parametrized family of maps on the circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit nonuniformly expanding behavior. This implies the existence of "chaotic" dynamics in dissipative homoclinic tangles in periodically perturbed differential equations.