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Abstract <p> We study the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1116_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic dynamical system generated by the (2-2) rational function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1116_Article_IEq3.gif" Format="GIF" Height="41" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(x)=\dfrac{ax^2}{1-x^2}\)</EquationSource> </InlineEquation>, with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1116_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">\(a,x\in \mathbb{Q}_p\)</EquationSource> </InlineEquation>, which has three different fixed points. The behavior of these points depends on some conditions over <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1116_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1116_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(|a|_p\)</EquationSource> </InlineEquation>, to be an attractor, indifferent or repellent point. We find these conditions, and also the basin of attraction and the Siegel disk, in each case. </p>

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Dynamical Systems of a (2-2)-Rational \(p\)-Adic Function with Three Fixed Points

  • 期刊论文

  • 发表日期：

    2025年5月13日

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  • J. Galeano-Peñaloza,
  • O. F. Casas-Sánchez

摘要

Abstract

We study the \(p\) -adic dynamical system generated by the (2-2) rational function \(f(x)=\dfrac{ax^2}{1-x^2}\) , with \(a,x\in \mathbb{Q}_p\) , which has three different fixed points. The behavior of these points depends on some conditions over \(p\) and \(|a|_p\) , to be an attractor, indifferent or repellent point. We find these conditions, and also the basin of attraction and the Siegel disk, in each case.