, Volume 318, Issue 3, pp 831-855

Ergodic Theory of Generic Continuous Maps

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Abstract

We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measures—a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps.

To further explore the mysterious behaviour of C 0 generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.

Communicated by G. Gallavotti
Partially supported by a CNPq/Brazil productivity-in-research (PQ) grant and by a FAPERJ/“Junior Rio de Janeiro State Scientist” fellowship.
With financial support from Fondation Sciences Mathèmatiques à Paris (France) and FAPERJ (Brazil).