The dependence of the rotation number on parameters in families of circle diffeomorphisms–often visualized as a “Devil’s Staircase”–is a fundamental object of study in one-dimensional dynamics. While the monotonicity and continuity of the rotation number is well-established, its regularity properties are more subtle. In this talk, we present Jacek Graczyk’s proof (1991) that for any C^2 one-parameter family of circle diffeomorphisms that also satisfy some other natural assumptions, the rotation number is Hölder continuous with exponent equal to ½. We will derive this result by first organizing frequency locking intervals according to the Farey tree, then establishing the universality of “harmonic scaling” in the parameter space with the help of this organization. 

We introduce a notion of generalized $(C, \lambda)$-structure for nonlinear diffeomorphisms of Banach spaces. The main differences to the classical notion of hyperbolicity are that we allow the hyperbolic splitting to be discontinuous and in invariance condition assume only inclusions instead of equalities for both stable and unstable subspace. These aspects allow us to cover Morse-Smale systems and generalized hyperbolicity.

We suggest that the generalized $(C, \lambda)$-structure for infinite-dimensional dynamics plays a role similar to ``Axiom A and strong transversality condition'' for dynamics on compact manifolds. For diffeomorphisms of reflexive Banach space we showed that generalized $(C, \lambda)$-structure implies Lipschitz (periodic) shadowing and is robust under $C^1$-small perturbations. Assuming that generalized $(C, \lambda)$-structure is continuous for diffeomorphisms of arbitrary Banach spaces we obtain a weak form of structural stability: the diffeomorphism is semiconjugated from both sides with any $C^1$-small perturbation.

This is an introductory talk around the general theme of Schrödinger operators and their transfer matrices. I will start discussing on the relation between density of states for discrete Schrödinger operators on Z and the rotation number, and we will see what happens when one studies a finite width band instead: symplectic operators and Maslov indices.

Gap labeling theorems connect the spectra of Schrödinger operators to the dynamics that generate their potentials. We will focus on the main ideas in R. Johnson’s approach to gap labeling, which uses the Schwartzman group. After showing how this group can be computed in several dynamical settings, I will present results (joint with D. Damanik and J. Fillman) that describe how these dynamical invariants determine, and in some cases ensure, the opening of spectral gaps. In particular, for potentials generated by the full shift on finitely many symbols, every label predicted by the gap labeling theorem is realized at large coupling. We will conclude with a discussion of the proof and in which scenarios certain labels fail to occur.

We present joint work with Artur Avila on delocalizing Schr\"odinger operators in arbitrary dimensions via arbitrarily small perturbations of the potential.