Groups & Dynamics

Given distinct hyperbolic structures m and m' on a closed orientable surface, how many closed curves have m- and m'-length roughly equal to x, as x gets large? Schwartz and Sharp's correlation theorem answers this question. Their explicit asymptotic formula involves a term exp(Mx) and 0<M<1 is the correlation number of the hyperbolic structures m and m'. In this talk, we will show that the correlation number can decay to zero as we vary m and m', answering a question of Schwartz and Sharp. Then, we discuss extensions of this correlation theorem to higher rank Teichmuller theory and find diverging sequences of SL(3,R)-Hitchin representations along which the correlation number stays uniformly bounded away from zero. This will motivate the introduction and study of the correlation number for Hitchin representation of punctured surfaces. This talk is based on joint work with Xian Dai and joint work with Nyima Kao.