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<p>This paper establishes a full equivalence between the open set condition and the positivity of pseudo-Hausdorff measure for the graph-directed attractor <i>K</i> generated by a self-affine graph-directed system of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1940_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="186" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{\phi _e(x)=A^{-1}(x+d_e)\}_{e\in {{\mathcal {E}}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>ϕ</mi> <mi>e</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>A</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>d</mi> <mi>e</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>e</mi> <mo>∈</mo> <mi mathvariant="script">E</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. We further give the precise value of the pseudo-Hausdorff measure of <i>K</i> through upper convex <i>s</i>-density of specific Markov measures, extending the previous results on self-affine sets to more complex self-affine graph-directed systems.</p>

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On the open set condition of self-affine graph-directed systems

  • 期刊论文

  • 发表日期：

    2025年7月24日

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  • Si-Wei Yue,
  • Lian Wang,
  • Dong-Hong Xiong

摘要

This paper establishes a full equivalence between the open set condition and the positivity of pseudo-Hausdorff measure for the graph-directed attractor K generated by a self-affine graph-directed system of the form \(\{\phi _e(x)=A^{-1}(x+d_e)\}_{e\in {{\mathcal {E}}}}\) { ϕ e ( x ) = A - 1 ( x + d e ) } e ∈ E . We further give the precise value of the pseudo-Hausdorff measure of K through upper convex s-density of specific Markov measures, extending the previous results on self-affine sets to more complex self-affine graph-directed systems.