An inequality in real Milnor-Thurston monotonicity problem

TL;DR

This paper extends a known inequality related to topological entropy monotonicity from quadratic to more general real polynomial families with rational exponents, using algebraic methods.

Contribution

It introduces a weak analog of Tsujii's inequality for the family $f_{c,r}(x)=|x|^r+c$ with rational $r>1$, broadening the scope of previous results.

Findings

Established a new inequality for the family $f_{c,r}$

Demonstrated algebraic approach applicability to non-integer exponents

Extended monotonicity analysis to a broader class of functions

Abstract

In late 1990's Tsujii proved monotonicity of topological entropy of real quadratic family $f_c(x)=x^2+c$ on parameter $c$ by proving an inequality concerning orbital information of the critical point. In this paper, we consider a weak analog of such inequality for the general family $f_{c,r}(x)=|x|^r+c$ with rational $r>1$, by following an algebraic approach.