Entropy, Lyapunov exponents, and rigidity of group actions

Abstract

Rudolph and Katok-Spatzier measure rigidity theorems 2.1 Furstenberg's conjecture; theorems byRudolph and Katok-Spatzier Furstenberg conjectureLet S 1 " R{Z be the additive circle. Note that for k P t2, 3, 4, . . . u the map M k : x Þ Ñ kx mod 1

Zimmer's conjecture for non-uniform lattices and escape of mass

“…As the arguments in this paper use in an essential way all arguments from [16] and many of those from [17], the reader may find it easier to read those papers first. An expository account of some of the arguments from [16] with more detailed background may be found in the lecture notes by Brown [14].…”

“…We may thus assume there is a i such that the restriction A 0 | ai fails to have len Γ -subexponential growth. Replacing a i with a −1 i if needed, a standard exercise (see for example [14,Proposition 3.1.2]) yields an a i -invariant, Borel probability measure µ 0 on X 0 such that λ top,ai,µ0,A0 > 0.…”

The disintegration of measures with periodic repetitive pattern

“…Measure disintegration. As we are exclusively working on M = Σ k × S 1 and the foliation considered is {{ξ} × S 1 } ξ∈Σ k we state Rohklin's disintegration theorem already for this context (see [12] or the appendix on measure disintegration of [8] for more general statements).…”

“…Just to mention a couple of them, it goes from the classical work of Anosov [1] that established ergodicity of conservative Anosov diffeomorphisms, for which the disintegration of volume along the stable and unstable foliations was needed to be understood, to the seminal work of Ledrappier and Young [15,16] which establishes and compile many links between disintegration of measures and entropy. See also the book [8].…”