Thin annuli property and exponential distribution of return times for weakly Markov systems

We deal with the problem of asymptotic distribution of first return times to shrinking balls under iteration generated by a large general class of dynamical systems called weakly Markov. Our ultimate result is that these distributions converge to the exponential law when the balls shrink to points. We apply this result to many classes of smooth dynamical systems that include conformal iterated function systems, rational functions on the Riemann sphere (C) over cap, and transcendental meromorphic functions on C. We also apply them to expanding repellers and holomorphic endomorphisms of complex projective spaces. One of the key ingredients in our approach is to solve the well known problem of appropriately estimating the measures of a large class of geometric annuli. This problem is differently referred to by different authors; we call it the Thick Thin Annuli Property. Having established this property, we prove that for nonconformal systems the aforementioned distributions converge to the exponential law along sets of radii whose relative Lebesgue measure converges fast to 1. In the context of conformal iterated function systems, we also establish the Full Thin Annuli Property, which gives the same estimates for all radii. In this way, we solve a long standing problem. As a result, we prove that the convergence to the exponential law holds along all radii for essentially all conformal iterated function systems and, with the help of the techniques of first return maps, for all the aforementioned conformal dynamical systems.