**Mahan Mj**

TIFR

April 27, 2017

**Motions of limit sets**:
Let $
ho_n : pi_1(S) o PSL_2(mathbb{C})$ be a sequence of discrete, faithful representations converging to a representation $
ho : pi_1(S) o PSL_2(mathbb{C})$. We study the question: Does the dynamics of $
ho_n(pi_1(S))$ on the Riemann sphere converge to that of $
ho(pi_1(S))$? We focus on the locus of chaotic dynamics, also called the limit set.
It is well known, thanks to celebrated work of Mane-Sad-Sullivan that for a parametrized family of quasifuchsian groups (or equivalently convex cocompact discrete surface subgroups of PSl(2,C)) the limit set moves holomorphically on the Riemann sphere. We shall discuss what happens when a sequence of such groups converges to a non-quasifuchsian group. A discontinuity phenomenon was discovered in joint work with Caroline Series. In further work with Ken'ichi Ohshika, we characterize when precisely these discontinuities arise.