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<p>In this note, we generalise Bourgain’s construction of finitely-supported symmetric measures whose Furstenberg measure has a smooth density from the case of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_999_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{SL}_2(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to that of a general simple Lie group. The proof is the same as Bourgain’s, except that the use of Fourier series is replaced by harmonic analysis on a maximal compact subgroup.</p>

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Absolutely continuous Furstenberg measures for finitely-supported random walks

  • 期刊论文

  • 发表日期：

    2025年3月28日

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  • Félix Lequen

摘要

In this note, we generalise Bourgain’s construction of finitely-supported symmetric measures whose Furstenberg measure has a smooth density from the case of \(\textrm{SL}_2(\mathbb {R})\) SL 2 ( R ) to that of a general simple Lie group. The proof is the same as Bourgain’s, except that the use of Fourier series is replaced by harmonic analysis on a maximal compact subgroup.