Duality in parameter space and approximation of measures for mixing repellers

Abstract

For one-dimensional expanding mapsT with an invariant measureμ we consider, in a parameter space, the envelope ℰ _n of the real lines associated to any couple of points of the orbit, connected byn iterations ofT. If the map hass inverses and is piecewise linear, then the sets ℰ _n are just the union ofs ⁿpoints and converge to the invariant Cantor set ofT. A correspondence between all the sets and their measures is established and allows one to associate the atomic measure on ℰ₁ to the completly continuous measure on the Cantor set. If the map is nonlinear, hyperbolic, and hass inverses, the sets ℰ _n are homeomorphic to the Cantor set; they converge to the Cantor set ofT and their measures converge to the measure of the Cantor set whenn→∞. The correspondence between the sets ℰ _n allows one to define converging approximation schemes for the map an its measure: one replaces each of thes ⁿdisjoint sets with a point in a convenient neighborhood and a probability equal to its measure and transforms it back to the original set ℰ₁. All the approximations with linear Cantor systems previously proposed are recovered, the converging proprties being straightforward in the present scheme. Moreover, extensions to higher dimensionality and to nondisconnected repellers arte possible and are briefly examined.