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Counter-examples to an infinitesimal version of the Furstenberg conjecture. (English) Zbl 1377.37067

Furstenberg’s \((\times 2, \times 3)\)-conjecture states that the Lebesgue measure \(\lambda\) on \(\mathbb{T}\) is the only non-atomic measure that is \((\times 2,\times 3)\)-invariant and ergodic [H. Furstenberg, Math. Syst. Theory 1 (1967), 1–49]. Furstenberg’s conjecture is a strong rigidity statement for higher-rank group actions.
The author formulates several versions of the above conjecture from the differential geometry point of view after endowing the probability measure space \(\mathcal{P}(\mathbb{T})\) with a differential structure [B. Kloeckner, Ergod. Th. & Dynam. Sys. 33 (2013), 529–548]. Then he shows that an infinitesimal version of the above conjecture is false in a very strong way: the vector space of tangent vectors at \(\lambda \in \mathcal{P}(\mathbb{T})\), that are invariant under the tangent actions induced by \(\times 2\) and \(\times 3\), is of infinite dimension; moreover, the vector space corresponding to the actions of \(\times d\)’s for all \(d\geq 2\) is of dimension two.
The author notices that the Lebesgue measure is indeed the only non-atomic measure that is invariant under \(\times d\)’s for all \(d\geq 2\).

MSC:

37E10 Dynamical systems involving maps of the circle
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
49Q20 Variational problems in a geometric measure-theoretic setting
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets

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References:

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