Exponential mixing for Gibbs measures on self-conformal sets and applications

In this paper, we show that Gibbs measures on self-conformal sets generated by a $C^{1+\alpha }$ conformal IFS on ${\mathbb{R}}^d$ satisfying the OSC are exponentially mixing. We exploit this to obtain essentially sharp asymptotic counting statements for the recurrent and the shrinking target subsets associated with any such set. In particular, we provide explicit examples of dynamical systems for which the recurrent sets exhibit (unexpected) behavior that is not present in the shrinking target setup. In the process of establishing our exponential mixing result we extend Mattila’s rigidity theorem for self-similar sets to self-conformal sets without any separation condition and for arbitrary Gibbs measures.