A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli

2018/03/05 by Gerber, Marlies, Kunde, Philipp
#37A35 #37C40 #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 37A20 #Secondary: 37A05

paper · doi:10.48550/arxiv.1803.01926

Abstract

Let M be a smooth compact connected manifold of dimension d≥ 2, possibly with boundary, that admits a smooth effective \mathbbT²-action S=\S_α,β\_(α,β) ∈ \mathbbT² preserving a smooth volume ν, and let B be the C^∞ closure of \h ∘ S_α,β ∘ h^-1 : h ∈ Diff^∞(M,ν), (α,β) ∈ \mathbbT²\. We construct a C^∞ diffeomorphism T ∈ B with topological entropy 0 such that T × T is loosely Bernoulli. Moreover, we show that the set of such T ∈ B contains a dense G_δ subset of B. The proofs are based on a two-dimensional version of the approximation-by-conjugation method.

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