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Attractors and time averages for random maps. (English) Zbl 0974.37036
A problem of interest is to know whether systems (diffeomorphisms) with infinitely many sinks or sources are negligible from the measure theoretical point of view. It has been conjectured that such systems correspond to zero Lebesgue measure in the parameter space, but this is not yet clarified. In the present paper, one shows that this conjecture is true in the setting of maps endowed with random noises. The basic result of the paper reads as follows: every diffeomorphism of a compact finite dimensional boundaryless manifold \(M\) under absolutely continuous random perturbations along a parametrized family has only finitely many physical measures of which basins cover Lebesgue’s – a.e. point of \(M\). One can conclude that there cannot exist infinitely many attractors of which orbits pass close to a quadratic homoclinic tangency point under random perturbation. Shortly, diffeomorphisms with infinitely many attractors are not stable under random perturbations.

MSC:
37H20 Bifurcation theory for random and stochastic dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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