Arithmeticity and topology of smooth actions of higher rank abelian\n groups

2013/05/30 by Anatole Katok, Federico Rodriguez Hertz, Katok, Anatole +1
Mathematics · #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.1305.7262

Abstract

We prove that any smooth action of mathbb Z^m-1, m\≥ 3 on an\nm-dimensional manifold that preserves a measure such that all non-identity\nelements of the suspension have positive entropy is essentially algebraic, i.e.\nisomorphic up to a finite permutation to an affine action on the torus or its\nfactor by \± Id. Furthermore this isomorphism has nice geometric properties,\nin particular, it is smooth in the sense of Whitney on a set whose complement\nhas arbitrary small measure. We further derive restrictions on topology of\nmanifolds that may admit such actions, for example, excluding spheres and\nobtaining below estimate on the first Betti number in the odd-dimensional case.\n

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