I will introduce the concept of weakly Markov dynamical systems. I will then consider the distributions of first return times to almost all shrinking balls for such systems. These distributions will be shown to converge to the exponential law along the radii of relative Lebesgue measure converging very fast to one.

The class of weakly Markov dynamical systems will be proved to include such large classes of smooth dynamical systems as expanding repellers, holomorphic endomorphisms of complex projective spaces, and Axiom A diffeomorphisms (none of them are assumed to be conformal) and conformal ones such as conformal iterated function systems, conformal graph directed Markov systems, conformal expanding repellers, rational functions of the Riemann sphere, and transcendental meromorphic functions.

For the whole class of conformal systems I will in fact prove more, namely that the convergence to the exponential law holds along all radii. This will be achieved by proving the thin annuli property.