Abstract <p> Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation> be a <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(2\)</EquationSource> </InlineEquation>-adic Ising mapping with parameter <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\in\mathbb Q_2\)</EquationSource> </InlineEquation>. The dynamics of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation> were investigated in the case <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\equiv3(\operatorname{mod }8)\)</EquationSource> </InlineEquation>. It was shown that, in this case, the set of bad points of the function (denoted by <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal P(f)\)</EquationSource> </InlineEquation>) consists of a single point, and that the study of the dynamical system can be restricted to the union of spheres <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq9.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bigsqcup_{n=3}^\infty S_{\frac{1}{2^n}}(1)\)</EquationSource> </InlineEquation>. It was proven that the Ising function acts as a bijective isometry on each sphere <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq10.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(S_{\frac{1}{2^n}}(1)\)</EquationSource> </InlineEquation>. Consequently, the function preserves the real-valued Haar measure on each such sphere. Moreover, necessary and sufficient conditions for the ergodicity of the function were established. In particular, it was shown that if <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\equiv19(\operatorname{mod }32)\)</EquationSource> </InlineEquation> (or <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\equiv3(\operatorname{mod }32)\)</EquationSource> </InlineEquation>), then for each <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq13.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(n>3\)</EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq14.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(n=3\)</EquationSource> </InlineEquation>), the sphere <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1131_Article_IEq10.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(S_{\frac{1}{2^n}}(1)\)</EquationSource> </InlineEquation> decomposes into two disjoint open compact minimal components. </p>

Ergodic Behavior of \((2,2)\)-Rational Ising Mapping over \(\mathbb Q_2\)

期刊论文
发表日期：

2025年11月10日

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Otabek Khakimov,
Khilola Yusupbaeva

摘要

Abstract

Let \(f\) be a \(2\) -adic Ising mapping with parameter \(a\in\mathbb Q_2\) . The dynamics of \(f\) were investigated in the case \(a\equiv3(\operatorname{mod }8)\) . It was shown that, in this case, the set of bad points of the function (denoted by \(\mathcal P(f)\) ) consists of a single point, and that the study of the dynamical system can be restricted to the union of spheres \(\bigsqcup_{n=3}^\infty S_{\frac{1}{2^n}}(1)\) . It was proven that the Ising function acts as a bijective isometry on each sphere \(S_{\frac{1}{2^n}}(1)\) . Consequently, the function preserves the real-valued Haar measure on each such sphere. Moreover, necessary and sufficient conditions for the ergodicity of the function were established. In particular, it was shown that if \(a\equiv19(\operatorname{mod }32)\) (or \(a\equiv3(\operatorname{mod }32)\) ), then for each \(n>3\) (resp. \(n=3\) ), the sphere \(S_{\frac{1}{2^n}}(1)\) decomposes into two disjoint open compact minimal components.