Let X be a compact complex manifold of dimension k, and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2106_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$f:X \longrightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⟶</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> be a dominating meromorphic map. We extend the concept of topological entropy by introducing the quantity <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2106_Article_IEq2.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$h_{(m,l)}^{top}(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="italic">top</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which measures the action of f on local analytic sets W of dimension l, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2106_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$W \subset f^{-n}(\Delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo>⊂</mo> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mi>n</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2106_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> is a local analytic set of dimension m. We then establish inequalities relating <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2106_Article_IEq2.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$h_{(m,l)}^{top}(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="italic">top</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to the Lyapounov exponents of suitable invariant measures.

Dimensional Entropy and Lyapounov Exponents for Meromorphic Maps

期刊论文
发表日期：

2025年7月12日

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Henry De Thélin

摘要

Let X be a compact complex manifold of dimension k, and $f:X \longrightarrow X$ $f : X ⟶ X$ be a dominating meromorphic map. We extend the concept of topological entropy by introducing the quantity $h_{(m,l)}^{top}(f)$ $h_{(m, l)}^{top} (f)$ , which measures the action of f on local analytic sets W of dimension l, where $W \subset f^{-n}(\Delta )$ $W \subset f^{- n} (Δ)$ and $\Delta $ $Δ$ is a local analytic set of dimension m. We then establish inequalities relating $h_{(m,l)}^{top}(f)$ $h_{(m, l)}^{top} (f)$ to the Lyapounov exponents of suitable invariant measures.