I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of $D$ has finitely many attractors whose union of basins of attraction has total probability;
The attractors of the elements in $D$ support a physical (SRB) measure"
I was wondering if there was some progress made in recent times. I am especially interested in continuous-time DS. I.e. could the conjecture be proven e.g. for continuous smooth DS on compact manifolds?
