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1 Chapter I. Continuous dynamical systems Let X be a metric space and R the additive group of the reals. Definition 1. A continuous dynamical system (R-action) on X is a continuous map ϕ : R × X → X satisfying the two conditions: 1. ϕ(0, x) = x for x ∈ X, 2. ϕ(t2, ϕ(t1, x)) = ϕ(t1 + t2, x) for x ∈ X and t1,t2 ∈ R. Definition 2. A reverse dynamical system to ϕ : R × X → X is the map −ϕ : R × X → X defined by ϕ(t, x) = ϕ(−t, x) For t ∈ R, the map ϕt : X → X is defined by ϕt (x) = ϕ(t, x). 1 Definition 3. Let x ∈ X. 1. The set γ(x) = ϕ(R × {x}) is the trajectory of x. 2. The set γ −(x) = {ϕ(t, x) |t ≤ 0} is the negative semi-trajectory of x. 3. The set γ +(x) = {ϕ(t, x) |t ≥ 0} is the positive semi-trajectory of x. A continuous dynamical system, particularly defined on a manifold, is often called a flow. Another name for a continuous dynamical system is directed 1-foliation. It is said that ϕ : R × X → X is defined on X, even though the domain of the function is R × X. A trajectory is often called an orbit. 2 Proposition 1. For every t ∈ R, ϕt is a homeomorphism. Proof. The maps ϕt and ϕ−t are continuous and inverse maps of each other: (ϕ−t ◦ ϕt )(x) = ϕ(−t, ϕ(t, x)) = ϕ(t − t, x) = ϕ(0, x) = x. Proposition 2. Let U ⊂ X be an open set. Then the set S V = x∈U γ(x) is open. Proof. V = x∈U γ(x) = {(t, x) | x ∈ U,t ∈ R} = S 3 S t∈R ϕt (U). Proposition 3. (G, ◦), where G = {ϕt |t ∈ R} and ϕs ◦ ϕt is the composition of maps, is an abelian group. Proof. Since ϕs ◦ ϕt = ϕs+t , the composition is well defined in G. Associativity holds in the composition of maps. The group identity is the identity map ϕ0. The inverse of ϕt is the map ϕ−t . We have (ϕ0 ◦ ϕt )(x) = (ϕt ◦ ϕ0)(x) = ϕt (x). G is abelian since (ϕs ◦ ϕt )(x) = ϕ(s, ϕ(t, x)) = ϕ(t + s, x) = ϕ(s + t, x) = ϕ(t, ϕ(s, x)) = (ϕt ◦ ϕs)(x). 4 Proposition 4. A trajectory is one of the following sets: • A one-point set (a singular point). The trajectory as well as the point are called singular. A singular point is also called a critical point, rest point, or fixed point. • A set homeomorphic to S1. Then the trajectory is called periodic or closed. • A one-to-one image of R. 5 Proof. Let x ∈ X. Suppose that γ(x) is not a one-to-one image of R, i.e., there are s0 and s00 in R, with s0 > s00 such that ϕ(s0, x) = ϕ(s00, x). Then ϕ(t, x) = x, where t = s0 − s00. Let t0 = inf{t > 0 | ϕ(t, x) = x}. Then either ϕ(t0, x) = x or there is a sequence (tn)tn>t0 convergent to t0 such that for each n ∈ N, ϕ(tn, x) = x. The continuity of ϕ implies that ϕ(t0, x) = x. Case 1. t0 > 0. For any t ∈ R, t = s + kt0, where s ∈ [0,t0] and k ∈ Z. The set {ϕ(t, x) |t ∈ [0,t0]} is homeomorphic to S1 and hence γ(x) is homeomorphic to S1. Case 2. t0 = 0. Then for any ε > 0, there is a tn such that ϕ(tn, x) = x and diam({ϕ(t, x) |t ∈ [0,tn]}) < ε. By an argument similar to the above, γ(x) = ({ϕ(t, x) |t ∈ [0,tn]}) and hence diam(γ(x)) = 0, so x is a singular point. 6 Definition 4. Let γ(x) be a periodic trajectory. If ϕ(t, x) = x and t > 0, then t is a period of γ(x). The number t0 = min{t ∈ R+ | ϕ(t, x) = x} is the fundamental period. Proposition 5. The numbers t and t0 in the above definition do not depend on the choice of the point of the trajectory. Example. Let S1 be the unit circle in the complex plane S1 = {(eθ i ∈ C | 0 ≤ θ < 2π}. Let ϕ : R × X → X be given by ϕ(t, eθ i) = e(at+θ )i, where a ∈ R+. Then the fundamental period if the trajectory is 2π a. 7 Proposition 6. The collection of all trajectories is a decomposition of X, i.e., X is the union of trajectories and trajectories are pairwise disjoint. Definition 5. • X and ϕ are called the phase space and the phase map, respectively. • The phase portrait is a geometric representation of the trajectories of a dynamical system in the phase space. • The directed phase portrait (in the phase space) is a geometric representation of the dynamical system restricted to individual trajectories. Note that a phase portrait for ϕ is also a phase portrait for −ϕ. 8 Figure 1: Brouwer “lines” Proofs Based on Geometric Intuition Krystyna Kuperberg There is no set L homeomorphic to a line that intersects every trajectory in one point. 9 Definition 6. Let ϕ : R × X → X and ψ : R × Y → Y on X be dynamical systems. If there is a homeomorphism h : X → Y such that the diagram idR ×h R×X → R ×Y ↓ϕ ↓ψ h X → Y commutes, then ϕ and ψ are (topologically) conjugate. Definition 7. Let ϕ : R × X → X and ψ : R × Y → Y on X be dynamical systems. If there is a homeomorphism h : X → Y such that for x ∈ X h(γ(x)) = γ(h(x)), then ϕ and ψ are topologically equivalent. 10 Proposition 7. If ϕ and ψ are conjugate, then they are topologically equivalent. Proof. fI ϕ and ψ are conjugate, then we have (t, x) 7→ (t, h(x)) ↓ ↓ ϕ(t, x) 7→ ψ(t, h(x)) Hence h(γ(x)) ⊂ γ(h(x)), the image of a trajectory is contained in a trajectory. Since Y decomposes into trajectories and h is onto, h(γ(x)) = γ(h(x)). 11 Example. Topological equivalence is a weaker condition than topological conjugacy. Let X = Y = S1 = {(eθ i ∈ C | 0 ≤ θ < 2π}, Define ϕ : R × X → X by ϕ(t, eθ i) = e(t+θ )i, and ψ : R × Y → Y by ϕ(t, eθ i) = e(2t+θ )i. ϕ and ψ are topologically equivalent but not topologically conjugate. 12 Definition 8. A set A ⊂ X is invariant if it is the union of trajectories. Proposition 8. If A and B are invariant then the following sets are invariant 1. B − A 2. the interior of A, Int(A), 3. the closure of A, Cl(A), 4. the boundary of A, Bd(A). Proof. Since the trajectories are pairwise disjoint, a trajectory is either contained in A or in B − A, thus B − A is invariant. Cl(A) is invariant since if y = limn→∞ xn, for some xn ∈ A, then for t ∈ R ϕ(t, y) = limn→∞ ϕ(t, xn). By Proposition 1, Int(A) is invariant, and hence Bd(A)= Cl(A)−Int(A) is invariant as well. 13 Proposition 9. If for λ ∈ Λ, the set Aλ is invariant then 1. the union A = S λ ∈Λ Aλ 2. the intersection A = is invariant, T λ ∈Λ Aλ is invariant. Definition 9. For x ∈ X define 1. The α-limit set of x, L−(x), as \ − Cl γ (ϕ(t, x)) t<0 2. The ω-limit set of x, L+(x), as \ + Cl γ (ϕ(t, x)) t>0 14 Proposition 10. − T 1. L (x) = n→∞ Cl (γ −(ϕ(tn, x))), where (tn) is any sequence of real numbers such that limn→∞ = −∞. + T 2. L (x) = n→∞ Cl (γ +(ϕ(tn, x))), where (tn) is any sequence of real numbers such that limn→∞ = ∞. The notation L−(x) and L+(x) is not standard. For example, α(x) and ω(x) are often used to denote the α-limit set and the ω-limit set, respectively. Proposition 11. If X is compact, then each of the limit sets L−(x) and L+(x) is compact and connected. 15 Lemma 1. The limit sets L−(x) and L+(x) are invariant. Proof. Consider the set L+(x). Let y ∈ L+(x) and let (tn), with limn→∞ tn = ∞, be a sequence such that limn→∞ ϕ(tn, x) = y. Then for t ∈ R, we have limn→∞ ϕ(t + tn, x) = limn→∞ ϕ(t, ϕ(tn, x)) = ϕ(t, y), by the continuity of φ , and clearly limn→∞ = ∞. Hence γ(y) ⊂ L+(x). A similar argument yields that L−(x) is invariant. The trajectory γ(x) may or may not be contained in L+(x) or L−(x). If it is contained in a limit set, then it is dense in this set. Hence if γ(x) is contained in each L+(x) and L−(x), then L+(x) = L−(x). Another interesting case of the limit sets being equal is when γ(x) is homoclinic. 16 Definition 10. A trajectory then γ(x) is homoclinic, if L+(x) = L−(x) and it is a singleton. 17 Definition 11. A set A ⊂ X is minimal if it is nonempty, compact, invariant, and minimal in this respect, i.e., if B ⊂ A is nonempty, compact, and invariant, then B = A. Proposition 12. A minimal set is the closure of a trajectory. The converse is not true. The closure of a trajectory need not be a minimal set. Definition 12. A dynamical system is minimal if X is a minimal set. Proposition 13. A dynamical system is minimal if every trajectory is dense. Definition 13. A trajectory γ(x) is ergodic, if Cl(γ(x)) = X. Definition 14. A dynamical system is transitive if there is an ergodic trajectory. 18 Proposition 14. If X is compact, then there is a minimal set. Proof. Let x ∈ X and let C = {Cλ }λ ∈Λ be the collection of all T compact, invariant sets containing x. The set λ ∈Λ Cλ is minimal. The minimal set is not unique. There may be many minimal sets. Every singular trajectory is a minimal set. Every periodic trajectory is a minimal set. 19