Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d’isométries des arbres

Media type: E-Article

Title: Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d’isométries des arbres

Contributor: Coornaert, M.; Papadopoulos, A.

imprint: American Mathematical Society (AMS), 1994

Language: English

DOI: 10.1090/s0002-9947-1994-1207579-2

ISSN: 1088-6850; 0002-9947

Keywords: Applied Mathematics ; General Mathematics

Origination:

Footnote:

Description: <p>Let <italic>X</italic> be a complete locally compact metric tree and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a group of isometries of <italic>X</italic> acting properly on this space. The space of bi-infinite geodesics in <italic>X</italic> constitutes a space <italic>GX</italic> on which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts properly. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the quotient of <italic>GX</italic> by this action. The geodesic flow associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the flow on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is the quotient of the geodesic flow on <italic>GX</italic>, defined by the time-shift on geodesics. To any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-conformal measure on the boundary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper X"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is an associated measure <italic>m</italic> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is invariant by the geodesic flow. We prove the following results: The geodesic flow on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Omega comma m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\Omega ,m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either conservative or dissipative. If it is conservative, then it is ergodic, If it is dissipative, then it is not ergodic unless it is measurably conjugate to the action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on itself by conjugation. We prove also a dichotomy in terms of the conical limit set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript c Baseline subset-of partial-differential upper X"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> <mml:mo>⊂</mml:mo> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Lambda _c} \subset \partial X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: the flow on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Omega comma m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\Omega ,m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is conservative if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-parenthesis normal upper Lamda Subscript c Baseline right-parenthesis equals mu left-parenthesis partial-differential upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu ({\Lambda _c}) = \mu (\partial X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and it is dissipative if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-parenthesis normal upper Lamda Subscript c Baseline right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu ({\Lambda _c}) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The results are analogous to results of E. Hopf and D. Sullivan in the case of Riemannian manifolds of constant negative curvature.</p>

Access State: Open Access

Search in field:

Recently searched for: