Equidistribution of small points, rational dynamics, and potential theory

If phi(z) is a rational function on P^1 of degree at least 2 with coefficients in a number field k, we compute the homogeneous transfinite diameter of the v-adic filled Julia sets of phi for all places v of k by introducing a new quantity called the homogeneous sectional capacity. In particular, we show that the product over all places of these homogeneous transfinite diameters is 1. We apply this product formula and some new potential-theoretic results concerning Green's functions on Riemann surfaces and Berkovich spaces to prove an adelic equidistribution theorem for dynamical systems on the projective line. This theorem, which generalizes the results of Baker-Hsia, says that for each place v of k, there is a canonical probability measure on the Berkovich space P^1_{Berk,v} over C_v such that if z_n is a sequence of algebraic points in P^1 whose canonical heights with respect to phi tend to zero, then the z_n's and their Galois conjugates are equidistributed with respect to mu_{phi,v} for all places v of k. For archimedean v, P^1_{Berk,v} is just the Riemann sphere, mu_{phi,v} is Lyubich's invariant measure, and our result is closely related to a theorem of Lyubich and Freire-Lopes-Mane.Comment: 50 pages; v2 contains additional references, exposition has been modified, and Sections 7 and 8 from v1 have been removed to shorten the paper's lengt

A four stage evolution of the White Channel gravel: Implications for stratigraphy and palaeoclimates

Although the White Channel gravel (WCG) of the Klondike district, Yukon, contains gold placers which have been exploited for over a century, few sedimentological studies have been undertaken. This study reports a four stage evolution of the WCG, comprising: i. An initial downcutting period which preferentially retained gold particles on the base of the strath. ii. An aggradational stage in which gold concentration occurred within sedimentary features. iii. A lacustrine layer representing a depositional hiatus. iv. A final, more rapidly aggrading fluvial stage. Identification of the lacustrine layer has clarified the evolution of the WCG depositional fluvial systems. Architectural element analysis and detailed sedimentological observations have been synthesized to gain a clearer understanding of the spatial variations within the WCG. Additionally, the identification of plant species from pollen within the lacustrine layer provides irrefutable evidence that the Klondike district was at least 7°C warmer during the Pliocene compared to the present

Anolis sheplani

Number of Pages: 2Integrative BiologyGeological Science