Hitchin discovered a component of the space of representations of a surface group into PSL(n,R), which bears many resemblances to the Teichmuller space of Fuchsian representations of the surface group into PSL(2,R). Labourie introduced dynamical techniques to show that these Hitchin representations are discrete, faithful quasi-isometric embeddings Sambarino associated Anosov flows to Hitchin representations whose periods record the spectral data of the representation.

In this talk, we will see how to use these flows to attach and study dynamical quantities to Hitchin representations, e.g. entropies, Liouville currents and associated Liouville volumes. We will also discuss rigidity results and natural Riemannian metrics on the Hitchin component. (These results are joint work with Martin Bridgeman, Francois Labourie and Andres Sambarino.)