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Ergodic Theory, Geometry and Dynamics Ergodic theory, geometry and dynamics C. McMullen 24 December, 2020 1 Contents 1 Introduction . 3 2 Ergodic theory . 12 3 Measures and topological dynamics . 35 4 Geometry of the hyperbolic plane . 46 5 Hyperbolic surfaces . 63 6 Arithmetic hyperbolic manifolds . 73 7 Dynamics on hyperbolic surfaces . 82 8 Orbit counting and equidistribution . 87 9 Double cosets and geometric configurations . 93 10 Unitary representations of simple Lie groups . 101 11 Dynamics on moduli space . 107 12 Unique ergodicity of the horocycle flow . 110 13 Ratner's theorem and the Oppenheim conjecture . 116 14 Geodesic planes in hyperbolic 3{manifolds . 120 15 Amenability . 126 16 Expanding graphs . 132 17 The Laplacian . 140 18 All unitary representations of PSL2(R) . 152 19 Expanding graphs and property T . 155 20 The circle at infinity and Mostow rigidity . 163 21 Ergodic theory at infinity of hyperbolic manifolds . 171 22 Lattices: Dimension 1 . 173 23 Dimension 2 . 176 24 Lattices, norms and totally real fields. 188 25 Dimension 3 . 191 26 Dimension 4, 5, 6 . 198 27 Higher rank dynamics on the circle . 199 28 The discriminant{regulator paradox . 202 29 Problems . 209 A Appendix: The spectral theorem . 231 1 Introduction These notes address topics in geometry and dynamics, and make contact with some related results in number theory and Lie groups as well. The basic setting for dynamics is a bijective map T : X ! X. One wishes to study the behavior of orbits x; T (x);T 2(x);::: from a topological or measurable perspective. In the former cases T is required to be at least a homeomorphism, and in the latter case one usually requires that T preserves a probability measure m on X. This means: m(T −1(E)) = m(E) (1.1) for any measurable set E, or equivalently Z Z f = f ◦ T for any f 2 L1(X; m). More generally one can consider a flow T : R × X ! X or the action of a topological group T : G × X ! X. In topological dynamics, one requires that T is a continuous map in the product topology; in measurable dynamics, that T is a measurable map. One can also consider the dynamics of an endomorphism T . Equation (1.1) defines the notion of measure preservation in this case as well. Ergodicity. A basic issue is whether or not a measurable dynamical system is `irreducible'. We say T is ergodic if whenever X is split into a disjoint union of measurable, T −1-invariant sets, X = A t B; either m(A) = 0 or m(B) = 0. Note: this definition makes sense for group actions too, and it makes sense even when m(X) is infinite, and when m is not preserved by T . Rotations. A simple and yet surprisingly rich family of examples of measure{ preserving maps is given by the rotations of the circle X = S1 = R=Z, define by T (x) = x + a mod 1: Theorem 1.1 The rotation T is ergodic if and only if a is irrational. 3 Proof. If a = p=q is rational then T has order q, Y = S1=hT i is a circle of length 1=q, and any measurable subset of Y pulls back to give a T {invariant measurable subset of S1. Otherwise, T has infinite order (and Y is the `tiny circle', inaccessible by traditional measure theory and topology, but amenable to the methods of non{commutative geometry). In this case, it is easy to see that every orbit [x] = (T ix : i 2 Z) is dense in S1. Otherwise, T would have to permute the intervals forming the complement of the orbit closure, preserving their length, which quickly implies that T is periodic. Now suppose A ⊂ S1 is T {invariant and m(A) > 0. Then, by the Lebesgue density theorem, for any > 0 we can find an interval I such that m(I \ A)=m(I) > 1 − . Since the orbit of I covers the circle, we con- clude that m(A) > 1 − . Hence m(A) = 1 and we have ergodicity. We will examine the irrational rotation from other perspectives in x2. Breadth of the topic. To indicate the range of topics related to ergodic theory, we now turn to some examples and applications. Examples of measure-preserving dynamical systems. 1. Endomorphism of S1. For S1 = R=Z, the group endomorphisms T (x) = dx, d 6= 0, are also measure preserving. 2. Blaschke products on the circle. A typical proper holomorphic map Q B : ∆ ! ∆ with B(0) = 0 has the form B(z) = z (z − ai)=(1 − aiz). This map preserves the linear measure m = jdzj=2π on the unit circle S1. For the proof, observe that the measure of a set E ⊂ S1 is given by uE(0), where uE(z) is the harmonic extension of the indicator function of E to the disk. Since B(0) = 0, we have −1 m(B )(E) = uB−1E(0) = uE ◦ B(0) = uE(0) = m(E): Sn 3. Interval exchange transformations. Let I = [0; 1] = 1 Ii be a tiling of the unit interval, and let f : I ! I be the map obtained by reordering these intervals. On each interval we have f(x) = x + ti for some i, and so linear measure is preserved. The case n = 2 is essentially the same as a rotation of S1. 4 4. Dynamics on the torus. The torus X = Rn=Zn also admits many interesting measure{preserving maps. In addition to the translations T (x) = x + a, each T 2 GLn(Z) determines a measure{preserving automorphism of X. In fact any matrix T 2 Mn(Z) with nonzero determinant gives a measure{preserving endomorphism of the torus. 5. Continued fractions. The continued fraction map x 7! f1=xg on [0; 1] preserves the Gauss measure m = dx=(1 + x) (with total mass log 2). This (together with ergodicity) allows us to describe the behavior of the continued fraction of a typical real number. 6. Stationary stochastic processes. A typical example is a sequence of inde- pendent, equally distributed random variables X1;X2;:::. The model Z+ Q space is then X = R with the product measure m = mi, with all the factors mi the same probability measure on R. The relevant dynamics then comes from the shift map, σ(X1;X2;:::) = (X2;X3;:::); which is measure{preserving. In this setting the Kolmogorov 0=1 law states than any tail event E has probability zero or one. A tail event is one which is independent of Xi for each individual i; for example, the event \Xi > 0 for infinitely many i" is a tail event. The zero{one law is a manifestation of ergodicity of the system (X; σ). 7. The H´enonmap. The Jacobian determinant J(x; y) = det DT (x; y) of any polynomial automorphism T : C2 ! C2, is constant. Indeed, J(x; y) is itself a polynomial, which will vanish at some point unless it is constant, and vanishing would contradict invertibility. (The Jacobian conjecture asserts the converse: a polynomial map T : Cn ! Cn with constant Jacobian has a polynomial inverse. It is know for polynomials of degree 2 in n variables, and for polynomials of degree ≤ 100 in 2 variables.) If T is defined over R, and det DT = 1, then T gives an area{preserving map of the plane. In the H´enonfamily H(x; y) = (x2 + c − ay; x), measure is preserved when a = 1. 5 Z 8. The full shift Σd = (Z=d) . 9. The baker's transformation. This is the discontinuous map given in base two by f(0:x1x2 :::; 0:y1y2 :::) = (0:x2x3 :::; 0:x1y1y2 :::): This map acts on the unit square X = [0; 1] × [0; 1] by first cutting X into two vertical rectangles, A and B, then stacking them (with B on top) and finally flattening them to obtain a square again. It is measurably conjugate to the full 2-shift. 10. Hamiltonian flows. Let (M 2n;!) be a symplectic manifold. Then any smooth function H : M 2n ! R gives rise to a natural vector field X, characterized by dH = iX (!) or equivalently !(X; Y ) = YH for every vector field Y . In the case of classical mechanics, M 2n is the n P cotangent bundle to a manifold N and ! = dpi ^dqi is the canonical symplectic form in position { momentum coordinates. Then H gives the energy of the system at each point in phase space, and X = XH gives its time evolution. The flow generated by X preserves both H (energy is conserved) and !. To see this, note that LX (!) = diX ! + iX d! = d(dH) = 0 and XH = !(X; X) = 0: In particular, the volume form V = !n on M 2n is preserved by the flow. 2n When M is a K¨ahlermanifold, we can write XH = J(rH), where J 2 = −I. In particular, in the plane with ! = dx ^ dy, any function H determines an area{preserving flow along the level sets of H, by rotating rH by 90 degrees so it becomes parallel to the level sets. The flow accelerates when the level lines get closer together. 11. Area{preserving maps on surfaces. The dynamics of a Hamilton flow on a surface are very tame, since each level curve of H is invariant under the flow. (In particular, the flow is never ergodic with respect to area measure.) One says that such a system is completely integrable. General area{preserving maps on surfaces exhibit remarkable universal- ity, and many phenomena can be consistently observed (elliptic islands 6 and a stochastic sea), yet almost none of these phenomena has been mathematically justified.
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