Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-archimedean dynamics

We determine when the equidistribution property for possibly moving targets holds for a rational function of degree more than one on the projective line over an algebraically closed field of any characteristic and complete with respect to a non-trivial absolute value. This characterization could be useful in the positive characteristic case. Based on the variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to Favre and Rivera-Letelier, using a dynamical Diophantine approximation theorem by Silverman and by Szpiro--Tucker. We also give a proof of a general equidistribution theorem for possibly moving targets, which is due to Lyubich in the archimedean case and due to Favre and Rivera-Letelier for constant targets in the non-archimedean and any characteristic case and for moving targets in the non-archimedean and 0 characteristic case.Comment: 25 pages, no figures. (v2: a few minor modifications

Approximation of Lyapunov exponents in non-archimedean and complex dynamics

We give two kinds of approximation of Lyapunov exponents of rational functions of degree more than one on the projective line over more general fields than that of complex numbers.Comment: 7 pages, to appear in the conference proceedings "Proceedings of the 19th ICFIDCAA Hiroshima 2011

Quantitative approximations of the Lyapunov exponent of a rational function over valued fields

We establish a quantitative approximation formula of the Lyapunov exponent of a rational function of degree more than one over an algebraically closed field of characteristic $0$ that is complete with respect to a non-trivial and possibly non-archimedean absolute value, in terms of the multipliers of periodic points of the rational function. This quantifies both our former convergence result over general fields and the one-dimensional version of Berteloot--Dupont--Molino's one over archimedean fields.Comment: 17 pages. Theorem 1 is improve

Equidistribution of rational functions having a superattracting periodic point towards the activity current and the bifurcation current

We establish an approximation of the activity current $T_c$ in the parameter space of a holomorphic family $f$ of rational functions having a marked critical point $c$ by parameters for which $c$ is periodic under $f$, i.e., is a superattracting periodic point. This partly generalizes a Dujardin--Favre theorem for rational functions having preperiodic points, and refines a Bassanelli--Berteloot theorem on a similar approximation of the bifurcation current $T_f$ of the holomorphic family $f$. The proof is based on a dynamical counterpart of this approximation.Comment: 12 page