Speaker: Katrin Gelfert 
Affiliation: UF Rio de Janeiro, Brazil


Hyperbolic dynamical systems, by definition, present two complementary directions, one with uniform (stable) contraction and the other with uniform expansion (unstable). A famous and important example of these systems is the so-called Smale horseshoe. However, there are many systems where hyperbolicity is not satisfied. We are interested in dynamics where the effects of contractions and expansions overlap generating regions with neutral behavior. We present dynamical systems with a simple description as a skew-product of a horseshoe and two circle diffeomorphisms. Important examples are provided by the projective actions of SL(2,R) matrices. The most interesting case occurs when one matrix is hyperbolic (eigenvalues different from one) and the other is elliptic (eigenvalues of modulo one). In this model two related systems come together: a non-hyperbolic dynamical system (with stable, unstable, and central directions) and a product of matrices (called cocycle). The Lyapunov exponent (expansion rate) of the central direction of the dynamic system corresponds to the exponential growth of the norms of the products of these matrices. We want to describe the “spectrum” of such "exponents”. To do this, it is necessary to understand the underlying “thermodynamic formalism” and the structure of space of “ergodic measures”, where the occurrence of so-called non-hyperbolic measures is a key difficulty. We aim to discuss this scenario, presenting key concepts and ideas and some results.

About Maths Colloquium

The Mathematics Colloquium is directed at students and academics working in the fields of pure and applied mathematics, and statistics. 

We aim to present expository lectures that appeal to our wide audience.

Information for speakers

Information for speakers

Maths colloquia are usually held on Mondays, from 2pm to 3pm, in various locations at St Lucia.

Presentations are 50 minutes, plus five minutes for questions and discussion.

Available facilities include:

  • computer 
  • data projector
  • chalkboard or whiteboard

To avoid technical difficulties on the day, please contact us in advance of your presentation to discuss your requirements.


Physics Annexe (06)
Room 407 (and via Zoom: