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<p>The strip entropy is studied in this article. First, we prove that the strip entropy approximation is valid for some ray of Markov tree-shifts on Markov-Cayley trees. This result extends the previous result of [Petersen-Salama, Discrete &amp; Continuous Dynamical Systems, 2020] on the conventional 2 tree. Second, we obtain that the rates of convergence of tree-shift associated with a primitive matrix <i>A</i> on the <i>d</i> tree are linear (with respect to the number of points in transversal direction), which is reminiscent of the result on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2102_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> [Pavlov, The Annals of Probability, 2012]. Lastly, we prove that the rates of convergence of strip entropy approximation are closely related to the expanding constant of the Markov-Cayley trees.</p>

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The strip entropy approximation of Markov tree-shifts on Markov-Cayley trees

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  • 发表日期：

    2025年8月2日

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  • Jung-Chao Ban,
  • Guan-Yu Lai,
  • Cheng-Yu Tsai

摘要

The strip entropy is studied in this article. First, we prove that the strip entropy approximation is valid for some ray of Markov tree-shifts on Markov-Cayley trees. This result extends the previous result of [Petersen-Salama, Discrete & Continuous Dynamical Systems, 2020] on the conventional 2 tree. Second, we obtain that the rates of convergence of tree-shift associated with a primitive matrix A on the d tree are linear (with respect to the number of points in transversal direction), which is reminiscent of the result on \(\mathbb {Z}^2\) Z 2 [Pavlov, The Annals of Probability, 2012]. Lastly, we prove that the rates of convergence of strip entropy approximation are closely related to the expanding constant of the Markov-Cayley trees.