Hyperbolic Manifolds, Discrete Groups and Ergodic Theory
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چکیده
1 Ergodic theory References for this section: CFS]. 1. The basic setting of ergodic theory: a measure-preserving transformation T of a probability space (X; B; m). Usually we assume T is invertible. (More generally, measure-preserving means R f T = R f; equivalently, m(T ?1 (A)) = mA.) How many measure spaces are there? Standard Borel spaces: any Borel subset of a complete, separable metric space X is Borel isomorphic to either 0; 1] or to a countable set. Any Borel probability measure on 0; 1] without atoms is Borel isomorphic to Lebesgue measure. See Mac], Ma, Prop 12.6], Roy]. 2. Examples of measure-preserving dynamical systems. Rotations of S 1. Translations in Lie groups and their homogeneous spaces. Automorphisms and endomorphisms of tori. Stationary stochas-tic processes. The Henon map. The Jacobian determinant of any polynomial auto-morphism of the plane is a constant. (The Jacobian conjecture asserts the converse: a polynomial endomorphism with constant Jacobian is invertible.) In the Henon family H(x; y) = (x 2 + c ? ay; x), measure is preserved when a = 1. Continued fraction map x 7 ! f1=xg on 0; 1] preserves m = dx=(1 + x). Blaschke products on the circle, B(z) = z Q (z ?a i)=(1 ?a i z), preserve m = dd. The baker's transformation. The shift d. Hamiltonian ows. Any function on R 2 gives rise to an area-preserving ow. The geodesic ow on a Riemannian manifold preserves Liouville 1
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