<p>In this paper we study some aspects of thermodynamic formalism, more specifically topological pressure and, as a consequence, topological entropy for piecewise smooth vector fields, using topological conjugation with shift maps and the Perron-Frobenius Operator. Some relationships between entropy and Hausdorff dimensions are also investigated. As consequences of our results, we obtain planar piecewise smooth vector fields with topological entropy and Hausdorff dimension equal to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10442_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$\log \alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10442_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in (1, 2]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. We stress that, for the best of our knowledge, it is the first time in the literature where the topological pressure and the Hausdorff dimension of a piecewise smooth vector field is obtained. Moreover, the obtainment of topological entropy given equal to a real number <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10442_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$\log \alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>, for all <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10442_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in (1, 2]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, is an extension of previous results where the topological entropy of the planar piecewise smooth vector fields considered are <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10442_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$\log k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, for <i>k</i> a positive integer.</p>

Some Aspects of Thermodynamic Formalism of Piecewise Smooth Vector Fields

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发表日期：

2025年6月16日

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Marco Florentino,
Tiago Carvalho,
Jeferson Cassiano

摘要

In this paper we study some aspects of thermodynamic formalism, more specifically topological pressure and, as a consequence, topological entropy for piecewise smooth vector fields, using topological conjugation with shift maps and the Perron-Frobenius Operator. Some relationships between entropy and Hausdorff dimensions are also investigated. As consequences of our results, we obtain planar piecewise smooth vector fields with topological entropy and Hausdorff dimension equal to $\log \alpha $ $log α$ for all $\alpha \in (1, 2]$ $α \in (1, 2]$ . We stress that, for the best of our knowledge, it is the first time in the literature where the topological pressure and the Hausdorff dimension of a piecewise smooth vector field is obtained. Moreover, the obtainment of topological entropy given equal to a real number $\log \alpha $ $log α$ , for all $\alpha \in (1, 2]$ $α \in (1, 2]$ , is an extension of previous results where the topological entropy of the planar piecewise smooth vector fields considered are $\log k$ $log k$ , for k a positive integer.