Counter-examples to an infinitesimal version of the Furstenberg conjecture

In this note we observe that one of our main results in "Optimal transport and dynamics of circle expanding maps acting on measures" (Ergodic Theory and Dynamical Systems 2013) has an interesting consequence: an infinitesimal version of the Furstenberg conjecture is false in a very strong way. More precisely, we find deformations of the Lebesgue measure on the circle which are first-order invariant simultaneously for all integer multiplications modulo $1$. We also correct an error in a lemma of the mentioned article. Both the proof and the statement must be corrected, but the main results of the paper are not affected. The postprint version of "Optimal transport and dynamics of circle expanding maps acting on measures" is a consolidated version, containing all the present content. It is available on the arXiv: https://arxiv.org/abs/1006.4491