Asymmetric unimodal maps with non-universal period-doubling scaling laws

TL;DR

This paper studies a family of asymmetric unimodal maps, develops a renormalization theory revealing non-universal super-exponential scalings, and explores the properties of their attractors and conjugacies.

Contribution

It introduces a new class of asymmetric unimodal maps with non-universal scaling laws and establishes a renormalization framework for analyzing their complex dynamics.

Findings

Existence of a Feigenbaum-Coullet-Tresser 2^∞ map within the family.

Renormalization intervals scale super-exponentially and non-universally.

Conjugacy of attractors is smooth under certain invariant conditions.

Abstract

We consider a family of strongly-asymmetric unimodal maps $\{f_t\}_{t\in [0,1]}$ of the form $f_t=t\cdot f$ where $f\colon [0,1]\to [0,1]$ is unimodal, $f(0)=f(1)=0$, $f(c)=1$ is of the form and $$f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)& \mbox{ for }x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta) &\mbox{ for }x>c, \end{array}\right. $$ where we assume that $\beta>1$. We show that such a family contains a Feigenbaum-Coullet-Tresser $2^\infty$ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $2^\infty$ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum-Coullet-Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

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