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<p>For any Lie group <i>G</i> we introduce a renormalization map <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\cal R}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">R</mi> </mrow> </math></EquationSource> </InlineEquation> on the space of simple <i>G</i>-extensions of interval exchange transformations. Using <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\cal R}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">R</mi> </mrow> </math></EquationSource> </InlineEquation> we prove weak mixing and cohomological non-equivalence for typical simple compact <i>G</i>-extensions of IETs. This extends a well-known result of Avila and Forni for <i>G</i> = <i>U</i>(1) to any compact connected Lie group <i>G</i>.</p><p>It is also the first result on nonabelian extensions of interval exchange transformations.</p>

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Nonabelian Rauzy–Veech renormalization

  • 期刊论文

  • 发表日期：

    2025年6月29日

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  • Dmitri Scheglov

摘要

For any Lie group G we introduce a renormalization map \({\cal R}\) R on the space of simple G-extensions of interval exchange transformations. Using \({\cal R}\) R we prove weak mixing and cohomological non-equivalence for typical simple compact G-extensions of IETs. This extends a well-known result of Avila and Forni for G = U(1) to any compact connected Lie group G.

It is also the first result on nonabelian extensions of interval exchange transformations.