<p>In this paper, the rotational entropy <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3443_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$h_r(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is investigated for a random dynamical system <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3443_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> on the torus. The formula of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3443_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$h_r(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is obtained for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3443_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> which satisfies certain assumptions, and the lower and upper bounds of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3443_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$h_r(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are given for more general <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3443_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>. Several examples are presented to show that these results may not hold without the assumptions. This work can be seen as a random version of the previous work (Jiang et al. in J Differ Equ 379:862–883, 2024), in which the rotational entropy was introduced and investigated as a homotopy invariant for any torus map.</p>

Rotational Entropy for Random Torus Maps

期刊论文
发表日期：

2025年4月21日

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Weifeng Jiang,
Zhengxing Lian,
Yujun Zhu

摘要

In this paper, the rotational entropy $h_r(\varphi )$ $h_{r} (φ)$ is investigated for a random dynamical system $\varphi $ $φ$ on the torus. The formula of $h_r(\varphi )$ $h_{r} (φ)$ is obtained for $\varphi $ $φ$ which satisfies certain assumptions, and the lower and upper bounds of $h_r(\varphi )$ $h_{r} (φ)$ are given for more general $\varphi $ $φ$ . Several examples are presented to show that these results may not hold without the assumptions. This work can be seen as a random version of the previous work (Jiang et al. in J Differ Equ 379:862–883, 2024), in which the rotational entropy was introduced and investigated as a homotopy invariant for any torus map.