 
Summary: ROBUST TRANSITIVITY AND TOPOLOGICAL MIXING FOR
C1
FLOWS
FLAVIO ABDENUR, ARTUR AVILA AND JAIRO BOCHI
Abstract. We prove that nontrivial homoclinic classes of Crgeneric flows are topo
logically mixing. This implies that given a nontrivial C1robustly transitive set of
a vector field X, there is a C1perturbation Y of X such that the continuation Y of
is a topologically mixing set for Y . In particular, robustly transitive flows become
topologically mixing after C1perturbations. These results generalize a theorem by
Bowen on the basic sets of generic Axiom A flows. We also show that the set of flows
whose nontrivial homoclinic classes are topologically mixing is not open and dense,
in general.
2000 Mathematics Subject Classification: 37C20.
Key words: generic properties of flows, homoclinic classes, topological mixing.
1. Statement of the Results
Throughout this paper M denotes a compact ddimensional boundaryless manifold,
d 3, and Xr
(M) is the space of Cr
vector fields on M endowed with the usual Cr
topology, where r 1 . We shall prove that, generically (residually) in Xr
