Observable Dynamics and the Generic Coincidence of Milnor, Statistical, and Physical Attractors

We study observable dynamics of generic continuous interval maps.

We prove that, for a residual subset of $C([0,1])$, the global Milnor, statistical, and physical attractors coincide with the non-wandering set.

Thus, for a typical interval map, the asymptotic dynamics detected by Lebesgue-almost every initial condition recovers all recurrent dynamics.

The common attractor is a robust Cantor set of zero Hausdorff dimension.

Moreover, every invariant probability measure has a basin of zero Lebesgue measure.

We also present examples showing that, outside the generic setting, positive topological entropy may occur outside the global Milnor attractor, and the three attractors need not coincide.