<p>In late 1990’s, Tsujii proved monotonicity of topological entropy of real quadratic family <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f_c(x)=x^2+c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation> on parameter <i>c</i> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f_{c,r}(x)=|x|^r+c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> </msup> <mo>+</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation> with rational <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$r>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, by following an algebraic approach.</p>

An inequality in the real Milnor-Thurston monotonicity problem

期刊论文
发表日期：

2025年11月20日

查看全文

Ziyu Li,
Minyu Lu,
Tianyu Wang

摘要

In late 1990’s, Tsujii proved monotonicity of topological entropy of real quadratic family $f_c(x)=x^2+c$ $f_{c} (x) = x^{2} + c$ 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 $f_{c,r}(x)=|x|^r+c$ $f_{c, r} (x) = {| x |}^{r} + c$ with rational $r>1$ $r > 1$ , by following an algebraic approach.