DSNT Seminar: The dimension drop conjecture for analytic iterated function systems

Abstract: The Hausdorff dimension of self-similar sets and measures arising from iterated function systems (IFS) of similarities has been known since the 1980s under the assumption of strong separation conditions. This dimension value, the similarity dimension, also happens to be the correct value for typical systems but it has been a hard open problem to say when exactly the Hausdorff dimension is not equal to the similarity dimension and when a `dimension drop' occurs. A sufficient condition is that images of the maps in the IFS overlap exactly and the dimension drop conjecture asserts that this is the only way a dimension drop can occur. A few years ago, Hochman made significant progress towards the conjecture by introducing a mild condition known as the exponential separation condition (ESC), which is satisfied e.g. when all the parameters in the system are algebraic numbers. No such easy parametrisation exists for more general analytic iterated function systems on R and the work of Hochman does not generalise to these IFSs. In this talk, I will give the current state of the art for sets and measures in R arising from analytic IFSs and show that the dimension drop conjecture holds `typically' in a strong topological sense.

Based on joint work with Balazs Barany and Istvan Kolossvary.

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).