Lecture 1 Monday, January 14

LECTURE NOTES FOR MATH 7394 ERGODIC THEORY AND THERMODYNAMIC FORMALISM INSTRUCTOR: VAUGHN CLIMENHAGA Lecture 1 Monday, January 14 Scribe: Vaughn Climenhaga 1.1. From deterministic physics to stochastic outcomes Imagine rolling a pair of dice. This is a basic example of randomness. But dice fly and bounce according to the laws of physics, which are deterministic. So where does the randomness come from? One answer: we are completely ignorant of the initial conditions. If we knew the starting positions of the dice, and how hard and in which direction they are thrown, we could compute the outcome and predict what number will be rolled. So the randomness is only there because we have no initial information. Suppose we try to actually carry this out. We observe the initial conditions, then feed them into a computer that performs a simulation and gives us a prediction. What could go wrong? The model we use is incomplete; ignores wind, earthquakes, Godzilla, etc. The computer makes numerical errors when it rounds things off. Our initial observations contained a small measurement error. All of these are important to consider, but for now we ignore the first two of these and focus on the third. How large does the initial measurement error need to be before it gives a wrong prediction? Imagine a different system; we put a ball on a roof and predict in which direction it will roll down. If we are on the side of the roof, then a small error in the initial measurement leads to a small error in the prediction; a small displacement remains small. But if we are on the peak of the roof, a small error in the initial measurement can change which side of the roof the ball goes down; a small displacement can grow very large. Some systems, such as the dice or as it turns out, the weather have the property that small displacements can grow very large, not just for certain initial conditions, but for all initial conditions. This sensitive dependence on initial conditions is the source of stochastic (random) behavior in deterministic systems, popularly called chaos. 1

2 MATH 7394 VAUGHN CLIMENHAGA 1.2. Flows: continuous-time systems Let s introduce some notation to study a system that evolves in time according to a deterministic rule. Let X be the set of all possible states of the system called the phase space. We assume that X is equipped with a metric, so that d(x, y) is small if the two states x and y are similar to each other, and large if they are very different. Given x X and t R, let f t (x) denote the state of the system at time t if x is the initial state at time 0. We assume that the map f t : X X is continuous for all t. We further assume that the system is autonomous time-independent meaning that if start in state x at time 0, evolve to state y = f t (x) at time t, and then evolve to state f t+s (x) at time t + s, the result is the same as if we had started at state y at time 0 and then evolved to state f s (y) at time s. In other words, (1.1) f t+s (x) = f s (f t (x)) for all s, t R and x X. Definition 1.1. A one-parameter family of maps {f t : X X} t R satisfying (1.1) is called a flow. We often refer to a flow as a continuous-time dynamical system. Given x X, the set {f t (x) : t R} is the trajectory or orbit of x under the flow. Example 1.2. Let σ = 10, β = 8/3, and ρ = 28, and consider the following system of ODEs that give a (very simplified) model of atmospheric convection: (1.2) dx dt = σ(y x), dy dt = x(ρ z) y, dz dt = xy βz. The Lorenz flow f t on R 3 is the flow obtained by solving these ODEs; that is, if (x, y, z) R 3 represents the initial conditions of the ODE, then the solution at time t is given by f t (x, y, z) R 3. The animation at https://www.youtube.com/watch?v=fye4jkaxsfy shows how three very similar initial conditions eventually diverge under the action of the flow. Observe that (0, 0, 0) is a fixed point of the flow, since dx = dy = dz = 0 dt dt dt at this point. If you remember your ODE theory, you can determine the stability of this fixed point by taking the 3 3 matrix of partial derivatives and finding its eigenvalues and eigenvectors; it turns out to have a two-dimensional stable subspace and a one-dimensional unstable subspace. The trajectories diverge when they come close to this fixed point on different sides of the stable subspace, and then are driven away along the unstable subspace in different directions. Another nice animation (with many more points) is at https://www.youtube. com/watch?v=5xu-9d4ahvu. As this one proceeds, it looks like the distribution of the points is somehow stabilizing; describing this phenomenon precisely leads to the notion of a Sinai Ruelle Bowen (SRB) measure, which is one of the central objects of this course.

ERGODIC THEORY AND THERMODYNAMIC FORMALISM - LECTURE NOTES 3 Lecture 2 Wednesday, January 16 Scribe: Vaughn Climenhaga 2.1. Invariant measures To be a little more precise (but only a little), we will want to have not only a metric d on X, but also a measure µ. The idea is that µ should describe how likely we are to find the system in a given state; thus µ should be a probability measure, meaning that µ(x) = 1. Then given E X, there is a probability µ(e) that the system is in a state represented by a point in E. 1 Say that µ is flow-invariant if this probability does not change over time; that is, if for every t R, the probability that f t (x) E is the same as the probability that x E. The former is given by µ(f t E) and the latter by µ(e), so we see that µ is flow invariant if (2.1) µ(f t E) = µ(e) for all t R. Question 2.1. Do flow-invariant measures exist? How many of them are there? If there is more than one, which one should we use? In the second video above, suppose we write m for the measure that describes the initial distribution of points; then m f t is the measure describing the distribution of points at time t. Question 2.2. Do the measures m f t converge (in some sense) to a flowinvariant limiting measure µ as t? If so, (how) does the asymptotic measure µ depend on the initial measure m? Suppose we have a function ϕ: X R that represents an observation or measurement of the system. Then ϕ f t represents the same measurement made at time t, and {(X, µ, ϕ f t )} t R represents a continuous-time stochastic process. Because f is deterministic, there is a strong correlation between ϕ f s and ϕ f t when s t. Question 2.3. Do these correlations get smaller as s t? To what extent does this stochastic process really behave randomly? Does it satisfy familiar limit laws such as the law of large numbers and the central limit theorem? These three questions will remain with us throughout the course, and will motivate most of our work. Their answers will reveal the mechanisms by which deterministic systems display stochastic behavior; for some general overviews of this topic, see [Rue80, ER85, Rue89]. There are also some nice videos and discussions at http://www.chaos-math.org/en. 1 This finesses the point of which subsets of X are measurable; what is the σ-algebra on which µ is defined? Since X is a metric space it is natural to consider the Borel σ-algebra, and so we will always consider Borel measures.

4 MATH 7394 VAUGHN CLIMENHAGA 2.2. Maps: discrete-time systems For several reasons, 2 we will spend most of our time studying discrete-time dynamical systems given by a map f : X X; here the idea is that time moves in discrete increments, and if x represents the present state of the system, then f(x) represents the state at the next time increment. We write f 2 (x) = f(f(x)) = (f f)(x), and similarly f n+1 (x) = f(f n (x)) for all n N. Iterating f in this way gives the forward orbit of x (its future), and if f is a bijection then we can also iterate f 1 to obtain the backward orbit of x (its past). Given a flow {f t } t R, one can obtain a discrete-time system by fixing t and studying the iterates of the time-t map f t : X X. Another method for transforming a continuous-time system into a discrete-time system is to take some subset A X of codimension 1 that is transverse to the flow direction and let F : A A be the first return map associated to the flow; that is, given x A we write r(x) for the first positive time at which f r(x) (x) A (the return time to A), and then put F (x) = f r(x) (x). One can also go in the reverse direction. Given a discrete-time dynamical system F : X X and a roof function r : X (0, ), let X r = {(x, t) : x X, 0 t < r(x)}; the suspension of (X, F ) with roof function r is given by defining { (x, t + s) if t + s < r(x), f s (x, t) = (F (x), t + s r(x)) if r(x) t + s < r(x) + r(f (x)), and letting f s for larger values of s be given by composing these cases. In other words, we flow vertically until we reach the graph of the roof function, and then jump back to the base (reset t to 0), iterating F when we do so. Note that this is not yet a true flow because we have not defined f s for s < 0. For this we need the extra assumption that F is invertible, and then one can show that each f s is invertible for s > 0, so putting f s = fs 1 gives a flow. So far we have just treated X and X r as sets; in general we also want to treat them as metric spaces, since after all the notion of distance played a key role in our earlier motivating discussion. If X is a metric space and r is continuous, then X r carries a natural quotient topology by thinking of it as the quotient space X r = {(x, t) : x X, 0 t r(x)}/ where (x, r(x)) (F (x), 0). If you have not studied abstract point-set topology, you can also view X r as a metric space by equipping it with the Bowen Walters metric [BW72] as follows. (1) First define the metric when r 1 is constant. (a) A vertical step (x, s) (x, t) has length s t. (b) A horizontal step (x, t) (y, t) has length (1 t)d(x, y)+td(f x, F y). 2 Such as the fact that discrete-time stochastic processes are often easier to handle than their continuous-time counterparts.

ERGODIC THEORY AND THERMODYNAMIC FORMALISM - LECTURE NOTES 5 (c) A basic path γ from (x, s) to (y, t) is a sequence of vertical and horizontal steps starting at (x, s) and terminating at (y, t); its length l(γ) is the sum of the lengths of the individual steps. 3 (d) d 1 ((x, s), (y, t)) = inf γ l(γ), where the infimum is over all basic paths from (x, s) to (y, t). (2) In the more general case when r(x) is not constant, define d r ((x, s), (y, t)) = d 1 ((x, s/r(x)), (y, t/y(t)), where the points in the second expression are taken to lie in X 1. See [BW72, LS19] for a proof that this is a metric. For the time being, the main take-away from this discussion is that questions and results about flows can be translated into questions and results about maps, and vice versa. Thus as a general rule, the study of discrete-time systems and the study of continuous-time systems should lead to very similar theories, although one should not take this too far, as there are certain subtleties that arise in one case but not the other. In the present course, we will focus largely on the discrete-time case and study maps (with a few notable exceptions). Our most elementary example is the doubling map f : S 1 S 1, where S 1 C is the unit circle and f(z) = z 2, so that the circle is wrapped around itself twice by the action of f. We often think of f as a map on the unit interval; identifying z = e 2πix S 1 with x [0, 1], we write f(x) = 2x (mod 1). See https://www.youtube.com/watch?v=zbbuxhwu7ye for an indication of how the Lorenz flow (or something like it) is related to the doubling map. 3 Observe that (x, s) (x, 1) = (F x, 0) (F x, t) is a basic path with only vertical steps.

6 MATH 7394 VAUGHN CLIMENHAGA References [BW72] Rufus Bowen and Peter Walters, Expansive one-parameter flows, J. Differential Equations 12 (1972), 180 193. MR 0341451 [ER85] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57 (1985), no. 3, part 1, 617 656. MR 800052 [LS19] Yuri Lima and Omri Sarig, Symbolic dynamics for three-dimensional flows with positive topological entropy, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 1, 199 256. MR 3880208 [Rue89] David Ruelle, Chaotic evolution and strange attractors, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1989, The statistical analysis of time series for deterministic nonlinear systems, Notes prepared and with a foreword by Stefano Isola. MR 1017644 [Rue80], Strange attractors, Math. Intelligencer 2 (1979/80), no. 3, 126 137. MR 595080