Chapter 1 Continuous time dynamical systems: a quick journey In this course we saw so far only examples and dynamical properties of discrete time dynam- ical systems. In this section, we briefly mention some of the corresponding definitions for continuous time dynamical systems to show that the theory is very similar. Furthermore, in many situations, the study of a continuous time dynamical system can be reduced to the study of a discrete time dynamical system by considering the first return or Poincar´emap on a section. We will present here two examples of continuous dynamical systems, the linear flow on a torus and the billiard in a circle, that, by this procedure, can be reduced to a rotation on the circle. 1.1 Flows: definition, examples and ergodic properties The evolution of a continuous dynamical system is described by 1−parameter family of maps ft : X ! X where t 2 R should be thought as continuous time parameter and ft(x) is the position of the point x after time t. The family of maps fftgt2R should satisfy the properties of a flow: Definition 1.1.1. A 1-parameter family fftgt2R is called a flow if f0 is the identity map and for all t; s 2 R we have ft+s = ft ◦ fs, i.e. ft+s(x) = ft(fs(x)) = fs(ft(x)); for all x 2 X: A continuous time dynamical system ft : X ! X is a space X together with a flow fftgt2R. The orbit O+ (x) = ff (x); t ≥ 0g of the flow is also called the trajectory of the point x under ft t the flow. The equation of a flow simply says that the position of the point x after time t + s can be obtained flowing for time s from the point ft(x) reached by x after time t. Remark that each −1 ft, t 2 R, is invertible since it follows from the definition of flow that f−t = ft . Let us give some examples of flows. Example 1.1.1. [Linear Flow on T2] Let T2 = [0; 1]2= ∼ be the two-dimensional torus, that is the unit square with opposite sides identified by the relation (x; 0) ∼ (x; 1) and (0; y) ∼ (1; y) (which glues top and bottom and right and left sides respectively). Let θ 2 [0; π=2) be an angle. 1 MAT 733 Continuous Time Dynamical Systems θ 2 2 θ 2 2 The linear flow 't : T ! T in direction θ is the flow 't : T ! T is given by x(t) = x + t cos θ mod 1; 'θ(x; y) = (x(t); y(t)); where t y(t) = y + t sin θ mod 1; are solutions of the differential equations dx dt = cos θ dy dt = sin θ: θ 2 The point 't (x; y) is obtained by considering the point (x + t cos θ; y + t sin θ) 2 R which belongs to the line through (x; y) in direction θ and has distance t from the initial point (x; y) and projecting it on T2 by π : R2 ! T2 given by π(x; y) = (x mod 1; y mod 1). θ In other words, 't (x; y), as t ≥ 0 increases, moves along the line trough (x; y) in direction θ θ until 't (x; y) hits the boundary of the unit square. After the trajectory hits the boundary of the square, it continuous on the opposite identified side of the square (see Figure 1.1(a)). If we draw the trajectories on the surface of a doughnut (which is another way of representating T2), the trajectories are continuous and look as in Figure 1.1(b). x2 x1 y3 y3 y1 y1 y4 y4 y2 y2 x2 x1 (a) (b) θ Figure 1.1: A trajectory of the linear flow 't on the square (a) and on the doughnut (b). Example 1.1.2. [Planar billiards] Let D ⊂ R2 be a closed subset of R2, for example a rectangle, a polygon or a circle, or more in general, any D whose boundary is a smooth curve with finitely many corners, see Figure 1.2. Figure 1.2: Billiard tables. Consider a point x 2 D and a direction θ 2 [0; 2π). Imagine shooting a billiard ball from x in direction θ on an ideal billiard table which is frictionless. The ball will move with unit speed along a billiard trajectory which is a straight line in direction θ until it hits the 2 MAT 733 Continuous Time Dynamical Systems boundary of D. At the boundary, if there is no friction, there is an elastic collision, that is the trajectory is reflected according to the law of optics (an exampe is shown in Figure ??): Law of optics: the angle of incidence, that is the angle between the trajectory before the collision and the tangent line to the boundary, is equal to the angle of reflection, that is the angle between the trajectory after the collision and the tangent line to the boundary. Figure 1.3: A trajectory of the billiard flow in a rectangle: elastic reflections according to the law of optics. If the trajectory hits a corner of the boundary, it ends there (as when a ball hit a pocket in a billiard table). In a real billiard game, one is interested in sending balls to a pocket. In a mathematical billiard, though, one considers only in trajectories which do not hit any pocket, and, since the billiard is frictionless, continue their motion forever. One is then interested in describing dynamical properties, for example asking whether trajectories are dense, or periodic, or trapped in certain regions of the billiard (see for example the circle billiard description in x 1.2.2 below). Let X = D × [0; 2π) be the set of pairs (x; θ) where x 2 D is a point in D and θ 2 [0; 2π) a 1 0 0 0 direction . The billiard flow bt : X ! X sends (x; θ) to the point bt(x; θ) = (x ; θ ), where x is the point reached after time t moving with constant unit speed along the billiard trajectory from x in direction θ and θ0 is the new direction after time t. The last example if the more general set up in which flows appear. Example 1.1.3. [Solutions of differential equations] Let X ⊂ Rn be a space, g : X ! Rn a function, x0 2 X an initial condition and x_(t) = g(x) (1.1) x(0) = x0 be a differential equation. If the solution x(x0; t) is well defined for all t and all initial conditions x0 2 X, if we set ft(x0) := x(x0; t) we have an example of a continuous dynamical system. In this case, an orbit is given by the trajectory described by the solution: Oft (x) := fx(x0; t); t 2 Rg: Exercise 1.1.1. Verify that ft(x0) := x(x0; t) defined as above satisfies the properties of a flow. Dynamical systems as actions A more formal way to define a dynamical system is the following, using the notion of action. Let X be a space and G group (as Z or R or Rd) or a semigroup (as N). 1To be more precise, one should consider as X the space of paris (x; θ) where if x belongs to the interior of D the angle can be any θ 2 [0; 2π), but if x belongs to the boundary of D, the direction θ is contrained to pointing inwards. 3 MAT 733 Continuous Time Dynamical Systems Definition 1.1.2. An action of G on X is a map : G × X ! X such that, if we write (g; x) = g(x) we have (1) If e id the identity element of G, e : X ! X is the identity map; 2 (2) For all g1; g2 2 G we have g1 ◦ g2 = g1g2 . A discrete dynamical systems is then defined as an action of the group Z or of the semigroup N. A continuous dynamical system is an action of R. There are more complicated dynamical systems defined for example by actions of other groups (for example Rd). Exercise 1.1.2. Prove that the iterates of a map f : X ! X give an action of N on X. The action N × X ! X is given by (n; x) ! f n(x): Prove that if f is invertible, one has an action of Z. Exercise 1.1.3. Prove that the solutions of a differential equation as (1.1) (assuming that for all points x0 2 X the solutions are defined for all times) give an action of R on X. Ergodic properties of flows The ergodic theory of continuous time dynamical systems can be developed similarly to the discrete time theory. The main ergodic theory definitions for flows are very similar to the corresponding ones for maps: Assume that (X; B; µ) is a measured space. Definition 1.1.3. A flow ft : X ! X is measurable if for any t 2 R, ft is a measur- able transformation and furthermore the map F : X × R ! X given by F (x; t) = ft(x) is measurable. A measurable flow ft : X ! X is measure-preserving and preserves the measure µ if −1 for any t 2 R, ft preserves µ, that is for any measurable set A 2 B, µ(ft (A)) = µ(A) transformation. Remark that since each transformation ft in a flow is invertible, one can equivalently ask that µ(ft(A)) = µ(A) for all A 2 B and t 2 R.