# Measure rigidity, quantum unique ergodicity, and the set of exceptions in Littlewood's Conjecture

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In 1967 Furstenberg discovered a very surprising phenomenon:

while both $T: x \to 2 x \bmod 1$ and $s: x \to 3 x \bmod 1$ on $\R / \Z$ have many closed invariant sets, closed sets which are invariant under both $T$ and $S$ are very rare (indeed, are either finite sets of rationals or $\R / \Z$). Furstenberg also conjectured that a similar result holds for invariant measures. This conjecture is of course still open.

As has been shown by several authors, including Katok-Spatzier and Margulis this phenomenon is not an isolated curiosity but rather a deep property of many natural $\Z ^ d$ and $\R ^ d$ actions ($d > 1$) with many applications.

Recently, there has been substantial progress in the study of measures invariant under such actions. While we are at present still far from full resolution of this intriguing question, the partial results we currently

have are already powerful enough to prove results in other fields. In particular, these techniques enable proving a special but important case of Rudnick and Sarnak's Quantum Unique Ergodicity Conjecture, as well as a partial result towards Littlewood's Conjecture on simultaneous diophantine approximations (the later is in a joint paper with M. Einsiedler and A. Katok).