EVENT DETAILS AND ABSTRACT

Title: A necessary condition for integrability and its implications
Speaker: Leo Butler
Speaker Info: Northwestern
Brief Description:
Special Note:
Abstract:

Benardete and Mitchell (1993) introduced a dynamical interpretation of Chen and Sullivan's construction of $\pi^1(M)$, the algebra of real-valued functions on $\pi_1(M)$. The former show that it is possible to define an asymptotic homotopy class of a curve that generalizes Schwartzman's definition of the asymptotic homology class. In this talk, I derive a couple necessary conditions for integrability of a flow and apply them to show the non-integrability of some geodesic flows on some nilmanifolds.

Let $\phi_t : M \to M$ be a $C^0$ flow, and let $\langle p \rangle$ denote the BM asymptotic homotopy class of the semi-orbit $\phi_t(p)$, $t \geq 0$.

Lemma: If $\phi_t$ is an integrable flow, then there exists an open dense subset $U \subset M$ such that for all $p \in U$, $\langle p \rangle$ exists.

Lemma: If $\phi_t$ is an integrable flow, then there exists an open dense subset $V \subset M$ such that if $p \in V$ then there exists a neighbourhood $V_p \subset V$ such that $q \in V_p$ implies $\langle p \rangle$ and $\langle q \rangle$ commute.

Theorem: There exists a family of compact nilmanifolds $\{N\}$ such that if $\phi_t : T^* N \to T^* N$ is geodesic flow induced by a left-invariant metric on the universal cover of $N$, then $\phi_t$ is non-integrable. In particular, there exists no open set $U \subset T^* N$ such that $U$ is homeomorphic to ${\bf T}^k \times {\bf R}^k$ and this homeomorphism conjugates $\phi_t$ to a translation-type flow.

This conclusion is surprising because the topological entropy of $\phi_t$ is zero.