Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. We observe how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the derormation space and of each stratum of the Teichmueller unit sphere bundle over the Riemann moduli space. We prove ergodicity for the analogous lift of the Weil-Petersson geodesic local. flow.Comment: 18 pages, no figures, presented at the Oxford conference honoring Nigel Hitchin's 70th birthday (9 September 2016) and to appear in the companion volume published by Oxford University Pres

C*-Algebras over Topological Spaces: The Bootstrap Class

We carefully define and study C*-algebras over topological spaces, possibly non-Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov theory over a topological space. We introduce and describe an analogue of the bootstrap class for C*-algebras over a finite topological space.Comment: Final version, very minor change

The primitive spectrum of a semigroup of Markov operators

For a semigroup S of Markov operators on a space of continuous functions, we use S-invariant ideals to describe qualitative properties of S such as mean ergodicity and the structure of its fixed space. For this purpose we focus on primitive S-ideals and endow the space of those ideals with an appropriate topology. This approach is inspired by the representation theory of C*-algebras and can be adapted to our dynamical setting. In the particularly important case of Koopman semigroups, we characterize the centers of attraction of the underlying dynamical system in terms of the invariant ideal structure of S

Discrete isometry groups of symmetric spaces

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $\Gamma<G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different characterizations of the subclass of Anosov subgroups, which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of various equivalent dynamical and geometric properties (such as asymptotically embedded, RCA, Morse, URU). 3. Discuss the topological dynamics of discrete subgroups $\Gamma$ on flag manifolds associated to $G$ and Finsler compactifications of associated symmetric spaces $X=G/K$. Find domains of proper discontinuity and use them to construct natural bordifications and compactifications of the locally symmetric spaces $X/\Gamma$.Comment: 77 page

Dynamical Systems in Categories

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We substantiate that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Preliminaries and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Preliminaries related to topology and measure theory . . . . . . . . 4 2.2 Basic notions from category theory . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Classical dynamical systems theory . . . . . . . . . . . . . . . . . . . . . . 23 3 Dynamical Systems in Abstract Categories . . . . . . . . . . . . . . . . . . 30 3.1 Monoids and monoid actions in abstract categories . . . . . . . . . . 31 3.2 Abstract dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Nonautonomous dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Dynamical Systems as Algebras and Coalgebras . . . . . . . . . . . . . .38 4.1 From monoids to monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 From abstract dynamical systems to monadic algebras . . . . . . . 48 4.3 Connections to coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4 Exponential objects in Top for locally compact Hausdorff spaces . . 52 4.5 (Co)Monadic (co)algebras and adjoint functors . . . . . . . . . . . . . .5