## CEMAPRE Seminar

### Dynamics of Flows with Expansive Measures

Keonhee Lee
(Chungnam National University, Daejeon Korea)

Abstract: We discuss some recent and ongoing works on the dynamics of flows with various expansive measures. In particular, we present a measurable version of the Smale's spectral decomposition theorem for flows. More precisely, we prove that if a flow $\phi$ on a compact metric space $X$ is invariantly measure expanding on its chain recurrent set $CR(\phi)$ and has the invariantly measure shadowing property on $CR(\phi)$ then $\phi$ has the spectral decomposition, i.e. the nonwandering set $\Omega(\phi)$ is decomposed by a disjoint union of finitely many invariant and closed sets on which $\phi$ is topologically transitive. Moreover we show that if $\phi$ is invariantly measure expanding on $CR(\phi)$ then it is invariantly measure expanding on $X$. Using this, we characterize the measure expanding flows on a compact $C^{\infty}$ manifold via the notion of $\Omega$-stability. {\it This is joint work with N. Nguyen.}

Thursday, July 4, 2019
Time: 10h30
Room: Sala Delta, Edificio Quelhas, ISEG
http://cemapre.iseg.ulisboa.pt/seminars/cemapre/