By Dave Witte Morris
The theorems of Berkeley mathematician Marina Ratner have guided key advances within the realizing of dynamical structures. Unipotent flows are well-behaved dynamical platforms, and Ratner has proven that the closure of each orbit for this kind of circulation is of an easy algebraic or geometric shape. In Ratner's Theorems on Unipotent Flows, Dave Witte Morris offers either an easy advent to those theorems and an account of the facts of Ratner's degree category theorem.
A selection of lecture notes geared toward graduate scholars, the 1st 4 chapters of Ratner's Theorems on Unipotent Flows should be learn independently. the 1st bankruptcy, meant for a reasonably common viewers, presents an creation with examples that illustrate the theorems, a few of their purposes, and the most principles enthusiastic about the evidence. within the following chapters, Morris introduces entropy, ergodic concept, and the speculation of algebraic teams. The booklet concludes with an explanation of the measure-theoretic model of Ratner's Theorem. With new fabric that hasn't ever prior to been released in e-book shape, Ratner's Theorems on Unipotent Flows is helping deliver those very important theorems to a broader mathematical readership.
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Extra resources for Ratner's theorems on unipotent flows
If µ ˆ is the product joining, or 1) µ ˆ has ﬁnite ﬁbers. 2) µ Proof. We omit some details (see Exer. 26 and Rem. 33). Consider two points (x, a) and (x, b) in the same vertical ﬁber. If the ﬁber is inﬁnite (and X is compact), then we may assume a ≈ b. 21), there is some t with at ≈ η1 (bt ). Let ξt be the vertical ﬂow on X × X, deﬁned by ξt (x, y) = x, ηt (y) . Then (x, a)t = (xt , at ) ≈ xt , η1 (bt ) = ξ1 (x, b)t . 5. 5F. Two points (x, a) and (x, b) on the same vertical ﬁber. We now consider two cases.
7(1). #10. Verify Eg. 7(2). #11. Give a short proof of Cor. 9, by using Ratner’s Measure Classiﬁcation Theorem. #12. Suppose Γ is a lattice in a Lie group G. Show that a subgroup Γ of Γ is a lattice if and only if Γ has ﬁnite index in Γ . #13. Suppose Γ and Γ are lattices in a Lie group G, such that Γ ⊂ Γ . Show that the natural map Γ \G → Γ \G has ﬁnite ﬁbers. #14. Verify Eq. 14). #15. Verify Eq. 16). #16. Suppose (ϕt , Ω ) is a quotient of a ﬂow (ϕt , Ω). Show that if ϕ is ergodic, then ϕ is ergodic.
10) Deﬁnition. ) if there is a conull subset X0 of X, such that ψ−1 (ω) ∩ X0 is ﬁnite, for all ω ∈ Ω. 11) Example. If Γ ⊂ Γ , then the natural quotient map ψ : X → X (cf. 7(1)) has ﬁnite ﬁbers (see Exer. 13). 12) Corollary (Ratner). ). In preparation for the direct proof of this result, let us develop some basic properties of unipotent ﬂows that are also used in the proofs of Ratner’s general theorems. Recall that ut = 1 t 0 1 and at = et 0 0 . e−t For convenience, let G = SL(2, R). 13) Deﬁnition.
Ratner's theorems on unipotent flows by Dave Witte Morris