Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$I=[0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$-1<\lambda <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>λ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$f:I\rightarrow I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> be a piecewise <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-affine map of the interval I. The exceptional set <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {E}_f$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">E</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> of f is the set of parameters <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$R_\delta \circ f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>δ</mi> </msub> <mo>∘</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> is not asymptotically periodic, where <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$R_\delta :I\rightarrow I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>δ</mi> </msub> <mo>:</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> is the rotation of angle <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>. In this paper we prove that <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {E}_f$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">E</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> has zero Hausdorff dimension. We derive this result from a more general theorem concerning piecewise Lipschitz contractions on <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1246_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> that has independent interest.

Hausdorff Dimension of the Exceptional Set of Interval Piecewise Affine Contractions

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2025年3月5日

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José Pedro Gaivão

摘要

Let $I=[0,1)$ $I = [0, 1)$ , $-1<\lambda <1$ $- 1 < λ < 1$ and $f:I\rightarrow I$ $f : I \to I$ be a piecewise $\lambda $ $λ$ -affine map of the interval I. The exceptional set $\mathcal {E}_f$ $E_{f}$ of f is the set of parameters $\delta \in \mathbb {R}$ $δ \in R$ such that $R_\delta \circ f$ $R_{δ} \circ f$ is not asymptotically periodic, where $R_\delta :I\rightarrow I$ $R_{δ} : I \to I$ is the rotation of angle $\delta $ $δ$ . In this paper we prove that $\mathcal {E}_f$ $E_{f}$ has zero Hausdorff dimension. We derive this result from a more general theorem concerning piecewise Lipschitz contractions on $\mathbb {R}$ $R$ that has independent interest.