Hyperbolicity of the invariant sets for the real polynomial maps

TL;DR

This paper investigates the hyperbolic nature of invariant sets for general real polynomial maps, extending known results from the logistic map to broader classes of polynomial functions.

Contribution

It explores the hyperbolicity of invariant sets in real polynomial maps with various zero configurations, broadening understanding beyond the logistic map case.

Findings

Hyperbolic behavior observed for certain parameter ranges

Invariant sets exhibit symbolic dynamics similar to logistic map

Results extend hyperbolicity concepts to more general polynomial maps

Abstract

It is well known that for $a>4$, the dynamical behaviors of the logistic map $f_a(x)=ax(1-x)$ on the maximal invariant compact set are "simple" which could be clearly explained by the theories of hyperbolic dynamics and symbolic dynamics. Is it possible that similar phenomena could be observed in general real polynomial maps? In this paper, we study this problem by investigating the real polynomial map $f_a(x)=ag(x)$, where $a$ is a parameter, and $g$ is a real-coefficient polynomial, which has at least two different real zeros or only one real zero.