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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.

The classical Morse theory proceeds by considering sublevel sets f^{−1}(−∞,a] of a Morse function f:M→R, where M is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets f^{−1}(a) and give conditions under which the topology of f^{−1}(a) changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse functions, the topology of a regular level f^{−1}(a) always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold M. When f is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the base space. (Counter-)examples and applications to celestial mechanics are also discussed.

Consider a system of periodic pendulum lattice with analytic weak coupling: i x ̈i +sinxi =−ε 􏰀 ∂xiβα(xj,xj+1,xj+2), xi =xi+N, i∈Z, j =i−2 where N 􏰃 3 is an integer, ε > 0 is a small parameter and the function βα is an analytic function of a certain form. It is shown in this paper that for small enough ε, the system admits motions such that the energy transfers between the pendulums in any predetermined order.