Horocycle flows on certain surfaces without conjugate points

Abstract

We study the topological but not ergodic properties of the horocycle flow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace h Subscript t Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {h_t}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the unit tangent bundle <italic>SM</italic> of a complete two dimensional Riemannian manifold <italic>M</italic> without conjugate points that satisfies the “uniform Visibility” axiom. This axiom is implied by the curvature condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K less-than-or-slanted-equals c greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>c</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K \leqslant c &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but is weaker so that regions of positive curvature may occur. Compactness is not assumed. The method is to relate the horocycle flow to the geodesic flow for which there exist useful techniques of study. The nonwandering set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h Baseline subset-of-or-equal-to upper S upper M"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mi>S</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h} \subseteq SM</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace h Subscript t Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {h_t}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is classified into four types depending upon the fundamental group of <italic>M</italic>. The extremes that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a minimal set for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace h Subscript t Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {h_t}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admit periodic orbits are related to the existence or nonexistence of compact “totally convex” sets in <italic>M</italic>. Periodic points are dense in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if they exist at all. The only compact minimal sets in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are periodic orbits if <italic>M</italic> is noncompact The flow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace h Subscript t Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {h_t}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is minimal in <italic>SM</italic> if and only if <italic>M</italic> is compact. In general <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace h Subscript t Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {h_t}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically transitive in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the vectors in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with dense orbits are classified. If the fundamental group of <italic>M</italic> is finitely generated and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h Baseline equals upper S upper M"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>S</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Omega _h} = SM</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace h Subscript t Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {h_t}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically mixing in <italic>SM</italic>.

Locations

• Transactions of the American Mathematical Society - View - PDF