| Résume | A doubling chart on an n-dimensional complex manifold Y is a univalent analytic mapping ψ : B_1 → Y of the unit ball in ℂ^n, which is extendible to the (say) four times larger concentric ball of B_1. A doubling covering of a compact set G in Y is its covering with images of doubling charts on Y. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney's ball coverings of a domain W ⊂ ℝ^n.
Our main motivation is that doubling coverings form a special class of “smooth parameterizations”, which are used in bounding entropy type invariants in smooth dynamics on one side, and in bounding density of rational points in diophantine geometry on the other. Complexity of smooth parameterizations is a key issue in some important open problems in both areas.
In this talk we present various estimates on the complexity of these objects. As a consequence, we obtain an upper bound on the Kobayashi distance in Y, and an upper bound for the constant in a doubling inequality for regular algebraic functions on Y. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on Y.
This is a joint work with Yosef Yomdin. |
