We consider iterated function systems (IFS) on the line, which are formed by a finite number of smooth contractions with non-vanishing derivatives. If the IFS is "non-overlapping" (the Open Set Condition holds), then the Hausdorff dimension of the attractor is the unique zero of the pressure function, sometimes called the "conformal dimension". The question is if/when there is a dimension drop in the "overlapping" case. For self-similar IFS significant progress has been made in the last ten years, starting with the work of M. Hochman (2014), but the non-linear case is much less understood.

I will present an example of an IFS for which two maps share a fixed point, but the dimension does not drop (this cannot happen for self-similar IFS). It is based on results from a joint work with Y. Takahashi (2021) on IFS of linear-fractional transformations; the latter builds on a joint work with M. Hochman (2017) on the Furstenberg measure for SL(2,R) random walks.