<p>This paper studies the hierarchical structure of periodic orbits of the automorphism induced by the matrix <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3403_Article_IEq1.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$A=\begin{pmatrix} 2& 1\\ 1& 1 \end{pmatrix}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> on the torus <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3403_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {T}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. The induced symbolic dynamics is not trivial with forbidden sequences. We show that the periodic orbits of the system is hierarchically structured by clusters. We establish the number of clusters via symbolic dynamics and digraphs. Algorithms that group all periodic orbits in clusters are given.</p>

Hierarchical Structure of Periodic Orbits of a Hyperbolic Automorphism on the 2-Torus

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发表日期：

2025年2月5日

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Huynh M. Hien,
Nguyen B. Tran,
Tran N. Nguyen

摘要

This paper studies the hierarchical structure of periodic orbits of the automorphism induced by the matrix $A=\begin{pmatrix} 2& 1\\ 1& 1 \end{pmatrix}$ $A = (\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix})$ on the torus ${\mathbb {T}}^2$ $T^{2}$ . The induced symbolic dynamics is not trivial with forbidden sequences. We show that the periodic orbits of the system is hierarchically structured by clusters. We establish the number of clusters via symbolic dynamics and digraphs. Algorithms that group all periodic orbits in clusters are given.