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$C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms
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abstract
In this paper, we establish the $C^0$-coerciveness of Moser's problem of mapping one smooth volume form to another in terms of the weak topology of measures associated to the volume forms. The proof relies on our analysis of Dacorogna-Moser's solution to Moser's problem of mapping one volume form to the other with the same total mass. As an application, we give a proof of smoothing result of area preserving homeomorphisms and its parametric version in two dimension, (or more generally in any dimension in which the smoothing theorem of homeomorphisms is possible, e.g., in dimension 3 but not necessarily in dimension 4). This in turn results in coincidence of the area-preserving homeomorphism group and the symplectic homeomorphism group in two dimension.
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Generic density of periodic orbits of area-preserving maps on punctured surfaces
C-infinity generic area-preserving diffeomorphisms on finitely punctured surfaces have dense periodic orbits, and on punctured spheres the orbits can be chosen equidistributed.
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