# Distribution of Orbits at Prime and Semi-prime Times

For a topological dynamical systems (T, X) and a fixed $x \in X$ we are interested in the distribution of prime and semi-prime orbits, i.e. ${T px}p='$ and ${T p1p2 x}p1,p2='$. We are interested in systems for which the related sequences of empirical measures

$$1/ \pi(N ) \Sum_p \leq N 1 \delta_T pX and 1/\pi_2 (N) \deltaT p1p2 x$$

have a limit for every $x \in X$ (we then say that (T, X) satisfies a PNT or SPNT respectively). It turns out that there are not so many examples of such systems. Our knowledge is especially limited for systems which have some non-trivial chaotic behavior, for example weak-mixing or mixing systems. I will discuss some known results and some recent progress. I will focus on examples of smooth and weakly mixing systems which satisfy a PNT and also on some mixing systems which satisfy a PNT. Finally I will discuss the case of SPNT for horocycle flows.