Dynamical Systems [Micha lMisiurewicz] Basic Notions, Examples The main objects studied in the theory of dynamical systems are maps f : X X, where X is a space with some structure. For instance, this structure can be topological,→ differentiable, or there may be a measure on X. Therefore three basic cases we will consider are: (a) X is a topological space (usually compact) and f is continuous, (b) X is a smooth (Cr, 1 r ) manifold (also usually compact) with or without boundary and f is smooth≤ (C≤r), ∞ (c) X is a space with a σ-field and a measure µ on it, the measure is a probability one, that is µ(X) = 1; f preservesF µ, that is µ(f −1(A)) = µ(A) for every A . Other way of saying that f preserves µ is that µ is f-invariant, or that f is an endomorphism∈ F of (X, ,µ). F 0 In each of cases (a)-(c), we consider the iterates of f, defined by induction: f = idX , and then f n+1 = f n f. Thus, f n = f f. This means that we look at the set of ◦ ◦···◦ n times n all iterates of f, f n∈Z+ . In other words, we look at the action of the semigroup Z+ on the space X.{ The} special case which| is{z considered} as often as the general one (and maybe even more often) is when the map f is invertible. We require then that f −1 is also within the class of maps we consider. That means that in case (a) f is a homeomorphism, in case (b) f is a diffeomorphism (of class Cr), and in case (c) f is an invertible measure preserving transformation (an automorphism) of X. If we consider an invertible f then we study also negative iterates of f, defined as f −n = (f −1)n = (f n)−1. Then we have the group Z acting on X. Yet another case is when we consider an action of the group R on X. This means that t instead of one map and its iterates we have a one-parameter group (ϕ )t∈R of invertible 0 s+t s t maps. That it is a group means that ϕ = idX and ϕ = ϕ ϕ for all s,t R. Usually t ◦ ∈ we call (ϕ )t∈R a flow. A classical example of such a situation is when X has a differentiable structure and there is a vector field F on X. Then we can consider the ordinary differential equationx ˙ = F (x). Under certain regularity conditions on F , the solutions to this equation are defined on the whole real line (the vector field is complete) and they depend in a smooth t way on initial conditions. If the solution with the initial condition x = x0 is (ϕ (x0))t∈R, t then (ϕ )t∈R is a one-parameter group of diffeomorphisms of X. In fact, this situation was the starting point of the whole theory of dynamical systems. Let us give several examples. Example 1. Let X = [0, 1] and let f : X X be given by the formula → 2x if 0 x 1/2, f(x) = 2 2x if 1/≤2 ≤x 1. − ≤ ≤ This is so called (full) tent map. The graph of this map is shown in Figure 1. Our space is a topological space – an interval, and the map is continuous. Therefore we have a situation from (a). We can also easily introduce an invariant measure. Let λ be the Lebesgue measure on [0, 1], and let be the σ-field of all measurable subsets of [0, 1]. If A then f −1(A) F ∈F 1 Dynamical Systems [Micha lMisiurewicz] is the union of two sets: one contained in [0, 1/2], the other in [1/2, 1], each of them of measure (1/2)λ(A) (see Figure 1). Hence, λ is f-invariant. Therefore we have a situation from (c). Notice that if B = f −1(A) [0, 1/2] then f(B) = A, but λ(B) = λ(A) (if λ(A) = 0). This shows that in the definition∩ of a measure preserving transformation6 we should6 really take the inverse image of A, not the image of A (of course, if we want this example to work). A f -1 ( A ) Figure 1 In fact, there is a deeper reason for taking the inverse image of A instead of the image of A. Namely, if g : X Y is a map and we have a measure µ on X then g carries this measure through to →Y and we get a measure ν = g∗(µ) on Y defined as follows. If A Y then we say that A is measurable if g−1(A) is measurable in X and then we set ν(A⊂) = µ(f −1(A)). It turns out that the family of measurable subsets of Y is a σ-field and ν is a measure. Exercise 1. Prove it. Using the above result, we see that f : X X preserves the measure µ if and only if f ∗(µ) = µ. → Example 2. Let F be a constant vector field in R2: F (x) = v for every x R2. The flow generated by this vector field is simply ϕt(x) = x + tv. Now consider the two-dimensional∈ torus T2 = R2/Z2. We can think of it as the square [0, 1]2 with opposite sides identified (upper with lower and left with right). Since F (x + k) = F (x) for every k Z2, the flow F induces a (constant) vector field on T2. Its flow is the projection of the flow∈ ϕt onto the torus. A piece of a trajectory of this flow is shown in Figure 2. Assume that v is not horizontal. If we take a horizontal circle S T2 and the first return map on S then we get a map f : S S. The system (S, f) is a⊂ so called Poincar´e t → section of the flow (ϕ )t∈R (see Figure 2). The first return map is defined as follows. Take a point of S and follow the flow until the trajectory hits again S – this point is the image of x. Since we can follow the trajectory back from f(x) to x, the map we get is invertible. 2 Dynamical Systems [Micha lMisiurewicz] If it happens, as in our example, that the first return map is defined on the whole S, it is a diffeomorphism. It is easy to show that in our example it is a rotation on the circle S. xf 3 (x ) f (x ) f 2 (x ) Figure 2 We now describe another basic example. Example 3. Let S be a finite set consisting of more than one point, for instance S = ∞ ∞ 1, 2,...,s with s > 1. Define Σ = −∞ S and Σ+ = S. More precisely, Σ = { ∞ } ∞ 0 −∞ Si and Σ+ = Si, where Si = S for each i. Thus, the elements of Σ are i= i=0 Q Q the doubly infinite sequences (...,x−2,x−1,x0,x1,x2,...) with xi S for all i, and the elementsQ of Σ are theQ usual one-sided sequences (x ,x ,x ,...) with∈ x S for all i. + 0 1 2 i ∈ We will regard Σ and Σ+ as topological spaces. The topology is defined as the product topology on Σ and Σ+, where S has the discrete topology. It can be described in three ways. The first way is to specify an open basis of the space. This is a family of open subsets such that a set A is open if and only if for every point x A there is a set B from the basis such that x B A. In our case an open basis we choose∈ will consist of all cylinders. ∈ ⊂ A cylinder is a set of the form Cy−n,y−n+1,...,yn−1,yn = (...,x−2,x−1,x0,x1,x2,...) Σ : x = y for all i n, n +1,...,n 1,n in{ the space Σ and of the form∈ i i ∈ {− − − }} Cy0,y1,...,yn = (x0,x1,x2,...) Σ : xi = yi for all i 0, 1,...,n in the space Σ+. Notice that the{ cylinders are not∈ only open, but also closed.∈ { }} The second way of describing topology on Σ and Σ+ is by specifying a metric. We do it by setting d(x,y)=2−k, where k is the smallest non-negative integer such that there is m with m = k and such that the m-th terms of the sequences x and y are different. | | (n) (n) (n) ∞ ∞ The third way is to say when limn→∞ x = y. If x =(xi )i=−∞ and y =(yi)i=−∞ (n) (n) ∞ ∞ (n) (for Σ) or x =(xi )i=0 and y =(yi)i=0 (for Σ+), then limn→∞ x = y if and only if (n) for every k there exists N such that if n N then xi = yi for every i with i k. This means that as n then x(n) coincides≥ with y on longer and longer pieces| around| ≤ 0. →∞ We define a shift σ on Σ and Σ+ (we will use the same letter in both cases) as the shift by one to the left. This means that σ(x0,x1,x2,...)=(x1,x2,x3,...). To write the formula for σ on Σ is more difficult. For this we have to introduce notation 3 Dynamical Systems [Micha lMisiurewicz] for the points of Σ which shows where the 0-th coordinate is.