Introduction to Ratner s Theorems on Unipotent Flows

Transcription

1 Introduction to Ratner s Theorems on Unipotent Flows Dave Witte Morris University of Lethbridge, Alberta, Canada dave.morris Dave.Morris@uleth.ca Part 2: Variations of Ratner s Theorem and Additional Ideas of the Proof Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

2 Ratner s Theorem [1991] G = Lie group = closed subgroup of SL(n, R) Γ = lattice in G X = G/Γ = discrete subgroup with vol(g/γ )< H = subgroup of G gen d by unips Hx = Sx for some subgroup S of G. Conjecture (Shah [1998]) Suffices to assume Zariski closure H gen d by unips smallest almost conn subgrp of G that H. Progress by Benoist-Quint [2013] True if H = SL(n, R) (or semisimple) and H/H is finitely gen d 0 Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

3 Ratner s Theorem G = Lie group = closed subgroup of SL(n, R) Γ = lattice in G X = G/Γ = discrete subgroup with vol(g/γ )< H = subgroup of G gen d by unips Hx = Sx for some subgroup S of G Derived from special case where H ={u t } is 1-dim l Ratner s Theorem {u t } unipotent {u t }x = Sx, subgrp S of G. Also: S {u t } and Sx has finite S-inv t volume. 0 Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

4 Example Let X = torus T 2 = R 2 /Z 2. Any v R 2 defines a flow on T 2 : x t = x + tv u t x If the slope of v is irrational, it is classical that every orbit is dense (& unif dist). Also: Lebesgue is the only inv t probability measure. v = (a, b, 0) defines a flow on T 3 = R 3 /Z 3 and T 2 {0} is invariant. Lebesgue on T 2 {0} is an invariant measure. v, any ergodic inv t probability meas for flow on T 3 is Lebesgue meas on some subtorus T k (0 k 3). Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

5 Ratner s Theorem on Orbit Closures {u t } unipotent {u t }x = Sx, subgrp S of G. Ratner s Equidistribution Theorem {u t }x is dense equidistributed in Sx: 1 T f (u t x) dt f d vol for f C c (G/Γ ) T 0 Sx where vol = S-invariant volume form on Sx. Ratner s Measure-Classification Theorem Any ergodic u t -invariant probability measure on G/Γ is S-invariant volume form on some Sx. Generalized to p-adic groups. [Ratner, Margulis-Tomanov] characteristic p? [Einsiedler, Ghosh, Mohammadi] measure-classification equidist orbit-closure Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

6 measure-classification equidistribution 1 T f (u t x) dt fdvol T Easy case 0 µ = unique u t -inv t probability meas on X (compact) every u t -orbit is equidistributed. Proof. M T (f ) := 1 T T 0 f (ut x) dt M T : C c (X) C positive linear functional Riesz Rep Thm: M T Meas(X) ={prob meas on X}. Banach-Alaoglu: Meas(X) weak compact subsequence M Tn M u t -invariant. Therefore M = µ. Sx So M T (f ) µ(f ) = f dµ. Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

7 Example: G = SL(2, R) Theorem (Furstenberg [1973]) SL(2, R)/Γ compact the only invariant probability measure for u t = 1 t 0 1 is the ordinary volume. Corollary Every u t -orbit is equidistributed ( dense). Theorem (Dani [1978]) SL(2, R)/Γ noncompact other ergodic u t -invariant measure is the measure supported on a closed orbit. Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

8 Theorem (Dani [1978]) SL(2, R)/Γ noncompact other ergodic u t -invariant measure is the measure supported on a closed orbit. General fact Every invariant measure is a combination of ergodic measures. Corollary Every u t -inv t probability meas on SL(2, R)/Γ is a combination of Lebesgue measure with measures on cylinders of closed orbits. Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

9 Ratner s Measure-Classification Theorem Ergodic u t -inv t prob meas is S-inv t vol on Sx ( S,x) Fix x 0. Sx 0 has finite S-inv t volume gsx 0 has finite gsg 1 -inv t volume. If u t gsg 1, this provides a u t -invariant measure. gsx 0 is analogue of a closed orbit. N(u t,s):= {g G u t gsg 1 } ( submanifold). N(u t, S)x 0 G/Γ cylinder of closed orbits: each gsx 0 has an u t -invariant prob meas. Lem. only countably many possible subgroups S. Cor. Every u t -inv t prob meas on G/Γ is a combo of measures on these countably many tubes. Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

10 What does ergodic mean? Defn. µ ergodic u t -inv t meas: every u t -inv t meas ble func is constant (a.e.) Pointwise Ergodic Theorem µ ergodic a.e. u t -orbit is µ-equidistributed. 1 T f (u t x) dt f dµ a.e. T 0 Exer. µ 1 µ 2, both ergodic µ 1 and µ 2 are mutually singular. µ 1 (C 1 ) = 1, µ 2 (C 2 ) = 1, C 1 C 2 = Hint. µ 1 = fµ 2 + µ sing and f is u t -inv t. X Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

11 Proof of Measure-Classification G = SL(2, R), u t = qx x u t qu t = 1 t e, a 0 1 s s 0 = 0 e s. u t x u tqx α + γt β + (δ α)t γt 2 γ δ γt Shearing: Fastest motion is parallel to the orbits. y x x t y t Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

12 Shearing Fastest motion is parallel to the orbits. y x x t y t α βe Contrast: a s qa s 2s = γe 2s δ Fastest motion is transverse to the orbits. y s y x x s Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

13 Shearing Fastest motion is parallel to the orbits. y x x t y t Key idea in proof of Measure-Classification Ignore motion along the orbit, and look at the transverse motion perpendicular to the orbit. y T 0 y x x t x T y t 0 y t Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

14 Example u t = 1 t 0 1, u t qu t = α + γt β + (δ α)t γt 2 γ δ γt Fastest motion is along {u t }. Ignoring this, largest terms are diagonal (in {a s }) Observation 1 a s a 0 1 s = 1 : a 0 1 s normalizes {u t }. Proposition For action of a unipotent subgroup, the fastest transverse divergence is along the normalizer. Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15

15 Prop. Fastest transverse div is along normalizer. Corollary (Step 1 of Ratner s Proof) µ is u t -inv t and ergodic (and... ) µ is a s -inv t. Proof. a s normalizes u t u t (a s µ) = a s (u t µ) = a s µ. µ and a s µ are two different ergodic measures live on disjoint u t -invariant sets C and a s C. Assume d(c, a s C) >ϵ. For x y in C: C u t x a s u t y a s C ( t,t ). d(c, a s C) <ϵ. Step 2: entropy calculation µ inv t under 1. 1 Dave Witte Morris (Univ. of Lethbridge) Introduction to Ratner s Theorems MSRI, Jan / 15