Displaying similar documents to “On singularly perturbed ordinary differential equations with measure-valued limits”

Most expanding maps have no absolutely continuous invariant measure

Anthony Quas (1999)

Studia Mathematica

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We consider the topological category of various subsets of the set of expanding maps from a manifold to itself, and show in particular that a generic C 1 expanding map of the circle has no absolutely continuous invariant probability measure. This is in contrast with the situation for C 2 or C 1 + ε expanding maps, for which it is known that there is always a unique absolutely continuous invariant probability measure.

The uniqueness of Haar measure and set theory

Piotr Zakrzewski (1997)

Colloquium Mathematicae

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Let G be a group of homeomorphisms of a nondiscrete, locally compact, σ-compact topological space X and suppose that a Haar measure on X exists: a regular Borel measure μ, positive on nonempty open sets, finite on compact sets and invariant under the homeomorphisms from G. Under some mild assumptions on G and X we prove that the measure completion of μ is the unique, up to a constant factor, nonzero, σ-finite, G-invariant measure defined on its domain iff μ is ergodic and the G-orbits...