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<p>We study the dynamics of generic volume-preserving automorphisms <i>f</i> of a Stein manifold <i>X</i> of dimension at least 2 with the volume density property. Amongst such <i>X</i> are all connected linear algebraic groups (except <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_177_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_177_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>) with a left- or right-invariant Haar form. We show that a generic <i>f</i> is chaotic and of infinite topological entropy, and that the transverse homoclinic points of each of its saddle periodic points are dense in <i>X</i>. We present analogous results with similar proofs in the non-conservative case. We also prove the Kupka–Smale theorem in the conservative setting.</p>

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Generic conservative dynamics on Stein manifolds with the volume density property

  • 期刊论文

  • 发表日期：

    2025年10月31日

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  • Leandro Arosio,
  • Finnur Lárusson

摘要

We study the dynamics of generic volume-preserving automorphisms f of a Stein manifold X of dimension at least 2 with the volume density property. Amongst such X are all connected linear algebraic groups (except \(\mathbb {C}\) C and \(\mathbb {C}^*\) C ∗ ) with a left- or right-invariant Haar form. We show that a generic f is chaotic and of infinite topological entropy, and that the transverse homoclinic points of each of its saddle periodic points are dense in X. We present analogous results with similar proofs in the non-conservative case. We also prove the Kupka–Smale theorem in the conservative setting.