Let f be a holomorphic automorphism of a compact Kähler manifold. Assume that f admits a unique maximal dynamic degree d_p with only one eigenvalue of maximal modulus. Let μ be its equilibrium measure. In this paper, we prove that μ is exponentially mixing for all d.s.h. test functions.

Let ψ be a conformal map on D with ψ(0)=0 and let F_α={z∈D:|ψ(z)|=α} for α>0. Denote by H^p(D) the classical Hardy space with exponent p>0 and by h(ψ) the Hardy number of ψ. Consider the limits L:=lim_{α→+∞} (log ω_D(0,Fα)^{−1} / logα), μ:=lim_{α→+∞}(d_D(0,Fα) / logα), where ω_D(0,Fα) denotes the harmonic measure at 0 of F_α and d_D(0,Fα) denotes the hyperbolic distance between 0 and F_α in D. We study a problem posed by P. Poggi-Corradini. What is the relation between L, μ and h(ψ)? Motivated by the result of Kim and Sugawa that h(ψ)=lim inf_{α→+∞} (log ω_D(0,Fα)^{−1} logα), we show that h(ψ)=lim inf_{α→+∞} (d_D(0,Fα) / logα). We also provide conditions for the existence of L and μ and for the equalities L=μ=h(ψ). Poggi-Corradini proved that ψ∉H^μ(D) for a wide class of conformal maps ψ. We present an example of ψ such that ψ∈H^μ(D).

We discuss the Kodaira problem for uniruled Kähler spaces. Building on a construction due to Voisin, we give an example of a uniruled Kähler space X such that every run of the K_X-MMP immediately terminates with a Mori fibre space, yet X does not admit an algebraic approximation. Our example also shows that for a Mori fibration, approximability of the base does not imply approximability of the total space.