In this talk, we will discuss the multifractal formalism for Bernoulli measures on self-affine carpets of Gatzouras–Lalley type. Along the way, we will discuss differences between “coarse” and “fine” multifractal theory, and understand why Gatzouras–Lalley measures can be badly behaved. In particular, the fine multifractal formalism can fail; the multifractal spectrum can even be discontinuous!
More generally I will also try to contextualize this project within a larger vision: to what extent can we understand the geometry of an invariant set using invariant measures supported on that set?
This is based on a joint work with Thomas Jordan and István Kolossváry.