---
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.'
address:
- 'Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, JAPAN'
- 'Department of Mathematics, University of Arizona, Tucson, AZ 85721'
author:
- Hiroki Takahasi and Qiudong Wang
title: Nonuniformly Expanding 1D Maps With Logarithmic Singularities
---

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Introduction
============

Let $f_{a}: {\mathbb R} \to {\mathbb R}$ be such that $$\label{f1-s1}
f_{a}\colon x\mapsto x+a+L\cdot\ln |\Phi(x)|, \ \  L>0,$$ where $a \in [0,1]$ and $\Phi\colon\mathbb R\to\mathbb R$ is $C^2$ satisfying: (i) $\Phi(x+1) = \Phi(x)$; (ii) $\Phi'(x)\neq0$ if $\Phi(x)=0$, (iii) $\Phi''(x)\neq0$ if $\Phi'(x)=0$. The family $(f_{a})$ induces a parametrized family of maps from $S^1={\mathbb R} /{\mathbb Z}$ to itself. In this paper we study the abundance of nonuniform hyperbolicity in this family of circle maps.

Our study of $(f_{a})$ is motivated by the recent studies of [@W; @WOk; @WOk11; @WO] on homoclinic tangles and strange attractors in periodically perturbed differential equations. When a homoclinic 