Rotation numbers of skew products and their dependence on a parameter
Abstract: Given a skew product over an ergodic transformation with the circle as the fiber, one can define the associated (fiberwise) rotation number. If the skew product depends continuously on a parameter, then so does the rotation number; my talk with be devoted to the regularity of this dependence.
This question is motivated, in particular, by the study of the discrete Schrödinger operator with dynamically defined potential. For such an operator, the distribution function of the density of states measure (DOS) is exactly the rotation number of the associated S^1-cocycle as a function of energy as a parameter.
I will present two results:
1) The rotation number is (under very mild assumptions) log-Hölder; this is a joint result with Anton Gorodetski (https://doi.org/10.1017/etds.2025.10195). This statement provides a dynamical viewpoint on the Craig-Simon theorem, stating the log-Hölder regularity for the DOS.
2) It turns out that the increment of the rotation number can be expressed in terms of invariant measures of the skew products (with the corresponding parameter values). For the random dynamics with i.i.d. maps, this gives a formula for the increment of the rotation number in terms of (forward and backward) stationary measures of the dynamics. This integral formula explains the analogy between known results on the regularity of DOS and stationary measures: the rotation number is at least as regular as the stationary measures (at least up to C^1 regularity). This is a joint work with Pedro Duarte and Anton Gorodetski (https://arxiv.org/abs/2512.00195).