January 2010 February 2010 March 2010 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 1 2 3 4 5 6 1 2 3 4 5 6 3 4 5 6 7 8 9 7 8 9 10 11 12 13 7 8 9 10 11 12 13 10 11 12 13 14 15 16 14 15 16 17 18 19 20 14 15 16 17 18 19 20 17 18 19 20 21 22 23 21 22 23 24 25 26 27 21 22 23 24 25 26 27 24 25 26 27 28 29 30 28 28 29 30 31 31 |

4:00 pm Friday, February 26, 2010 Geometry-Analysis: ABSOLUTELY CONTINUOUS INVARIANT MEASURES AND OTHER INVARIANT GEOMETRIC STRUCTURES FORby Anatole Katok (Penn State) in HB 227- We will discuss main steps in the proofs of some of the results described in the colloquium talk ``Dynamics, homotopy and rigidity''. Those results have been during the period from 2007 till 2010 jointly with Boris Kalinin and Federico Rodriguez Hertz in various combinations. We will first consider actions of Z^k on the torus T^{k+1}, k>2 that induce a hyperbolic action on the first homology group. Such an action is called homotopically Cartan. We show that it always preserves an absolutely continuous measure m and an invariant flat affine structure smooth in the Whintey sense defined on a set whose m-measure is arbitrary close to one Furthermore, there is a continuous factor-map (semi-conjugacy) between the action and an affine action that is bijective on a set of full measure and also on a large set of periodic points. The measure m is the unique measure that is mapped to Lebesgue measure by the semi-conjugacy. We will quickly mention recent extension of those results beyond the homotopically Cartan case to a more general class of Z^k, k>1 actions on a torus. Next we consider Z^k and R^k actions on k+1 and 2k+1 dimensional manifolds correspondingly, preserving a measure satisfying certain dynamical conditions, and show that the measure is absolutely continuous. While the conclusions are similar to those in homotopically Cartan case on the torus, absence of an a priori semi-conjugacy requires new powerful technical tools. Finally we consider an application of the results about actions of abelian groups to the Zimmer program. We prove that a real-analytic action of SL(n,Z) on the torus whose elements are homotopic to the elements of the standard linear action. We prove that such an action is analytically conjugate to the linear one on an open set that is mapped to the complement of finitely any periodic orbits of the linear action.
Submitted by dani@rice.edu |