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<p>We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, provides explicit examples of ergodic <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3246_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {R}}\)</EquationSource> </InlineEquation>-extensions of minimal locally Hamiltonian flows with non-degenerate saddles in genus two. More generally, given any symmetric irreducible permutation, we show that for almost every choice of lengths, the skew-product built over the associated IET using a cocycle with symmetric logarithmic singularities that is <i>odd</i> when restricted to each of the exchanged intervals is ergodic.</p>

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Ergodicity of explicit logarithmic cocycles over IETs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月4日

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  • P. Berk,
  • F. Trujillo,
  • C. Ulcigrai

摘要

We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, provides explicit examples of ergodic \({\mathbb {R}}\) -extensions of minimal locally Hamiltonian flows with non-degenerate saddles in genus two. More generally, given any symmetric irreducible permutation, we show that for almost every choice of lengths, the skew-product built over the associated IET using a cocycle with symmetric logarithmic singularities that is odd when restricted to each of the exchanged intervals is ergodic.