SYSTEMS. Russell Johnson

Transcription

1 NONAUTONOMOUS DYNAMICAL SYSTEMS Russell Johnson Università di Firenze Luca Zampogni Università di Perugia Thanks for collaboration to R. Fabbri, M. Franca, S. Novo, C. Núñez, R. Obaya, and many other colleagues. Dynamics and Differential Equations Dedicated to Prof. George R. Sell Minneapolis, 22 25/06 /

2 Introduction A nonautonomous dynamical system (NDS) can be of continuous or discrete type. Usually a continuous NDS is generated by the solutions of a nonautonomous differential equation x = f(t, x), t R, x B (1) where B is a Banach space. A discrete NDS is usually obtained by solving x n+1 = f n (x n ), n N, f n : B B. Let us consider the case (1) when B = R d. Suppose that each initial condition (t 0, x 0 ) gives rise to a unique global solution ϕ(t, t 0, x 0 ). For t R, set ˆτ t : R R d R R d : ˆτ t (t 0, x 0 ) = (t 0 +t, ϕ(t, t 0, x 0 )). Then {ˆτ t t R} is a one-parameter group of homeomorphisms, or dynamical system, on R R d. This is simple but already important. 2

3 If f satisfies recurrence conditions in t, one can often compactify time (Bebutov, Sell). That is, there exists a compact metric space Ω, a dynamical system {τ t } on Ω, and a function F : Ω R d R d such that the solutions ϕ(t, t 0, x 0 ) of x = F (τ t (ω), x) (1 ω ) define a dynamical system {ˆτ t } on Ω R d via ˆτ t (ω, x 0 ) = (τ t (ω), ϕ(t, ω, x 0 )). Moreover there is a point ω 0 Ω such that (1 ω0 ) coincides with (1 ω ). The upshot is that one can apply methods of dynamical systems theory to study the solutions of the family (1 ω ). 3

4 Topics of current interest include: Bifurcation Theory Oscillation Theory, Control Theory Spectral Theory of Ordinary Differential Equations Sturm-Liouville problems, Integrable systems Nonautonomous Functional Differential Equations Finite-Interval problems. 4

5 One needs a good linear theory to make inroads on these problems. In fact some are inherently linear, and one can study the others by the method of linearization along a solution. So, let us consider the linear nonautonomous system x = a(t)x, x R d. (2) Here a( ) takes values in the set M d of d d real matrices. Suppose that a( ) is bounded and uniformly continuous (for most purposes boundedness and measurability are enough). We assume that a( ) has some recurrence properties (e.g., almost periodicity, Birkhoff recurrence, chain recurrence,...). 5

6 One studies a constant coefficient system x = ax by making systematic use of the eigenvalues and generalized eigenspaces of a. Let us look for quantities which are related to the nonautonomous equation (2) and which are analogous to: (α) real parts of eigenvalues (β) generalized eigenspaces (γ) imaginary parts of eigenvalues. 6

7 Introduce the Bebutov hull (Ω, {τ t }) of a( ), and let A : Ω M d be the corresponding function. Consider the family x = A (τ t (ω)) x. (2 ω ) Let Φ ω (t) be the fundamental matrix solution of (2 ω ). One obtains a flow {ˆτ t } on Ω R d as follows: ˆτ t (ω, x) = (τ t (ω), Φ ω (t)x). This is a so-called skew-product flow. 7

8 (α) The notion of real part of an eigenvalue has two very useful generalizations to the nonautonomous case: Lyapunov exponents Dynamical (Sacker-Sell) spectrum. If ω Ω, the corresponding (maximal) Lyapunov exponent is β(ω) = lim sup t 1 t ln Φ ω(t). The existence of the limit and the properties of β( ) are studied in the Oseledets theory, of which more later. To define the dynamical spectrum, we introduce the concept of exponential dichotomy, or ED for short (Perron, Coppel, Palmer,...). 8

9 Definition 1 The family (2 ω ) admits an ED over Ω if there are constants K > 0, k > 0 and a continuous projection-valued function P : Ω M d : ω P ω = P 2 ω such that Φ ω (t)p ω Φ 1 ω (s) Ke k(t s), t s Φ ω (t)(i P ω )Φ 1 ω (s) Ke k(t s), t s. Thus an ED exists when the solution space Ω R d admits a hyperbolic splitting. Definition 2 A real number λ belongs to the dichotomy spectrum Σ of (2 ω ) if the translated family x = [ λi + A(τ t (ω))]x does not admit an ED over Ω. Clearly if A( ) reduces to a constant a, then Σ = {Rµ µ is an eigenvalue of a}. Sacker and Sell proved that, if (Ω, {τ t }) is chain recurrent, then Σ is a union of finitely many compact intervals. It is known that each Lyapunov exponent β(ω) lies in Σ. 9

10 (β) The concept of generalized eigenspace has two nonautonomous versions: the Oseledets bundles and the Sacker-Sell bundles (also Millionščikov and Selgrade). Oseledets Let µ be a {τ t }-ergodic measure on Ω (thus µ is indecomposable in a certain sense). Then Ω R d = W 1 W 2 W s where W 1,..., W s are µ-measurable, {ˆτ t }-invariant vector subbundles of Ω R d. Within each subbundle W i, all trajectories (τ t (ω), Φ ω (t)x 0 ) give rise to the same Lyapunov exponent (if x 0 0): β(ω, x 0 ) = β i = lim t ± 1 t ln Φ ω(t)x 0. 10

11 Sacker-Sell Let (Ω, {τ t }) be chain recurrent, and let Σ = [a 1, b 1 ]... [a r, b r ] be the spectrum. There is a decomposition Ω R d = W 1 W 2... W r where W 1,..., W r are continuous, {τ t }-invariant vector subbundles of Ω R d. For each i = 1, 2,..., r, the dynamical spectrum of ( W i, {ˆτ t }) is [a i, b i ]. An Oseledets bundles is always contained in a Sacker-Sell bundle; the reverse inclusion need not hold. 11

12 (γ) In the case of a constant-coefficient system, one views the imaginary part of an eigenvalue as the source of rotation of certain solutions of the system. This notion can be rendered concrete for certain types of families (2 ω ) via the concept of rotation number. Let us briefly discuss the rotation number in the context of two-dimensional families (2 ω ). Introduce polar coordinates r 2 = x x2 2, θ = arctan x 2 x 1 and write out the θ-equation θ = g(τ t (ω), θ) =... Definition 3 Let µ be an ergodic measure on Ω. The rotation number is α = lim t θ(t) t where θ(t) is the solution of the θ-equation. 12

13 It is known that the limit exists for µ-almost all ω Ω (it is easy to see that it is insensitive to the initial value θ(0)). One obtains a quantity which is very useful in the study of the spectral theory of ordinary differential operators, for example the Schrödinger operator (in one space dimension), Sturm-Liouville operators, the AKNS operator, and operators of Atkinson type. The initial work involved in applying the rotation number to these operators was carried out by J.-Moser, Kotani, and other scientists. A higher-dimensional version on the rotation number has been developed for linear Hamiltonian systems, and has been applied to spectral problems of Atkinson type and to control problems (J., J.-Nerurkar, Novo-Núñez- Obaya,... ). These matters have been studied by Fabbri, J., Novo, Núñez, Obaya; see the recent book of these authors. 13

14 Let us now consider an application of the basic tools of Nonautonomous Dynamics to the 1-D Schrödinger operator. We will discuss questions concerning the generalized reflectionless Schrödinger potentials (Lundina, Marchenko, Kotani). Let q : R R be a bounded continuous function. Introduce the differential expression L = d2 dx 2 + q(x). Then L determines an unbounded self-adjoint operator (Schrödinger operator) on L 2 (R). It also determines two half-line operators L ± defined on L 2 [0, ± ] respectively, via the Dirichlet boundary condition in x = 0: L ± ϕ = λϕ ϕ(0) = 0. 14

15 EXAMPLE. Consider the soliton potential q(x)=-2 d2 ln det(i + A(x)) (3) dx2 where A(x) = (A ij (x)) n i,j=1 and A ij (x) = li l j η i + η j e (η i+η j )x. Here l 1,..., l n and η 1,..., η n are positive numbers. If q(x) is of the form (3), then the spectrum of the operator L consists of the half-line [0, ) (where it is absolutely continuous) together with the finitely many eigenvalues η 2 1,..., η2 n.

16 A soliton potential has a very important property. Let g(x, y, λ) be the Green s function (integral kernel) of (L λ) 1 for λ in the resolvent of L. Then Rg(x, x, λ + i0) = 0 (λ > 0, x R), (4) where λ+i0 means the limit of g(x, x, λ+iε) as ε decreases to zero. One says that q (or L) is reflectionless. 15

17 To summarize: a soliton potential is reflectionless in (0, ), and has finitely many eigenvalues in (, 0). Let c < 0, and set { R c = cls q q is a soliton potential, and the eigenvalues of L = d2 + q(x) lie in (c, 0) dx2 Here the closure is taken in the topology of uniform convergence on compact subsets of R. One says that R c consists of generalized reflectionless Schrödinger potentials. Let GR = R c. c<0 }. Theorem (Lundina, 1985) The set R c is compact (in the compact-open topology). 16

18 There is a well-known connection between the Schrödinger operator L and the Korteweg-de Vries equation u t = 3u u x 1 3 u 2 x 3, u(0, x) = q(x). It was proved by Gardner-Green-Kruskal-Miura that a regular solution u(t, x) of the K-dV equation has the property that the family of Schrödinger operators L t = d2 + u(t, x) dx2 is isospectral (in L 2 (R)). This fact can be used to determine explicit solutions of the K- dv equation in certain cases (including that of soliton potentials). 17

19 Let q be a generalized reflectionless Schrödinger potential, that is q R c = GR. Marchenko c<0 tried to prove in 1991 that each such q gives rise to a regular solution of the K-dV equation. Along the way he worked out a nice parametrization of the elements in GR. About ten years later Kotani succeeded in proving that indeed each q GR gives rise to a solution u(t, x) of the K-dV equation, which in fact is meromorphic in the entire complex (t, x) space. He used the theory of the infinite-dimensional Grassmann-type space Gr 2, which is due to Sato-Segal-Wilson. 18

20 From now on we set c = 1 and unify R = R c = R 1. We will study certain compact subsets of R which are invariant under the natural translation flow, defined by τ x (q)( ) = q(x + ) (x R, q R). Let = a 0 < b 0 < a j < b j 0 (j 1), where b 0 1 and {(a j, b j ) j 1} are pairwise disjoint nonempty open intervals. Set then set E = R \ j=0 (a j, b j ), Q E = {q R L q = d2 dx 2 + q(x) has spectrum E and is reflectionless} By reflectionless we mean that the appropriate generalization of condition (2) holds, namely Rg q (x, x, λ + i0) = 0 (a.a λ E, all x R). (2 bis ) 19

21 Note: condition (2 bis ) holds on 0 < λ < for every q R. However if q R and L q has spectrum E, it need not be the case that (2 bis ) holds on the set E [ 1, 0]. Proposition Let E and Q E be as above. Suppose also that E has locally positive measure. Then Q E is compact and translation invariant. At this point the methods of Nonautonomous Dynamics become relevant. Introduce the set of divisors D E = {(y j, ε j ) y j [a j, b j ], ε j = ±1, 1 j < }. The points (a j, ±1) and (b j, ±1) are identified, so D E is a product of countably many circles. 20

22 Now we want to define a map π : Q E D E which will provide information about Q E and about the flow (Q E, {τ x }). To do this we need to carry out a preliminary discussion. Let q Q E. Consider the map g q : λ g g (0, 0, λ), which is defined and meromorphic on Ω E = C \ E. The map g q ( ) is strictly monotone increasing on each interval (a j, b j ). There are three cases to consider. (i) The map g q has a (unique) zero µ j (a j, b j ). (ii) The map g q is positive in (a j, b j ); in this case set µ j = a j. (iii) The map g q is negative in (a j, b j ); in this case set µ j = b j. 21

23 We define a divisor d q = {(µ 1, ε 1 ),..., (µ j, ε j ),... } D E where the signs ε j are determined using the Weyl m-functions m ± (λ). Recall that these functions are defined as follows: if λ C \ E, there are nonzero solutions ϕ ± (x, λ) L 2 [0, ± ) of Lϕ = λϕ. Set Then sign Im ±(λ) Iλ m ± (λ) = ϕ ± (0, λ) ϕ ± (0, λ). = ±1 if Iλ 0, and also sign dm ±(λ) = ±1 dλ if λ R \ E. (5) It turns out that 1 g q (λ) = m (λ) m + (λ). So if g q (λ) = 0 and a j < λ < b j then either m (λ) or m + (λ) has a pole at λ, and it follows from (5) that at most one of m ± (λ) has a pole at λ. 22

24 Now return to the definition of the divisor d q (the pole divisor). If µ j (a j, b j ), set ε j = 1 if m + ( ) has a pole at µ j, and ε j = 1 if m ( ) has a pole at µ j. If µ j = a j or µ j = b j then there is no need to worry about the sign ε j, since (a j, ±1) and (b j, ±1) are identified. So we obtain a map π : Q E D E : q d q = = {(µ j, ε j ) j 1}. We also obtain a pole motion : let τ x (q) = q(x+ ) be the translation, and set τ x (d q ) = {(µ j (x), ε j (x)) j 1} := d τx (q). It is not a priori clear that we obtain a flow on D E from this construction. This is because trajectories could conceivably cross, and in any case it is not clear that π is continuous. 23

25 The idea now is to produce examples via a study of the map π. The starting point is the following result. Theorem Let E R be a closed set of the form E = R \ j=0 (a j, b j ), where = a 0 < b 0 < a j < b j 0 (j 1), where b 0 1 and the closed intervals [a j, b j ] are pairwise disjoint. Suppose that E has locally positive measure. Then (a) The map π : Q E D E : q {(µ 1, ε 1 ),..., (µ j, ε j ),... } is continuous and surjective. (b) If the half-line operators L ± have purely absolutely continuous spectrum for all q Q E, then π is also injective, hence is a homeomorphism. 24

26 One can use this theorem to construct examples of sets E for which Q E has interesting structure. Example 1 There exists a closed set E R, which satisfies the hypotheses of the Theorem, such that Q E contains a minimal set M which is almost automorphic in the sense of Bochner-Veech, but is not Bohr almost periodic. We explain the terminology. A minimal flow {M, {τ x }) is Bohr almost periodic if there is a metric d on M, which is compatible with its topology, such that the flow {τ x } is isometric: d(τ x (q 1 ), τ x (q 2 )) = d(q 1, q 2 ) for all q 1, q 2 M and all x R. A minimal flow (M, {τ x }) is almost automorphic if there exists an almost periodic flow (M 0, {τ x }) and a flow homomorphism h : M M 0 such that h 1 (q 0 ) is a singleton for some q 0 M 0. The almost periodic minimal set of the example turns out to be the character group J E of the infinitely connected domain Ω E = C \ E. The domain Ω E is of Parreau-Widom (PW) type. 25

27 This example contradicts the Kotani-Last conjecture, according to which if the operators L ± all have purely a.c. spectrum for all q Q E, then Q E should consist entirely of Bohr almost periodic potentials. For other examples: Avila, Damanik-Yuditskii. 26

28 The Character Group of a Parreau-Widom domain The discussion is based on work by Sodin- Yuditskii, Volberg-Yuditskii. Let E = R \ j=0 (a j, b j ) where a 0 =, 1 b 0 < 0 and the nonempty open intervals (a j, b j ) are pairwise disjoint and contained in (b 0, 0). We define the character group J E of the domain Ω E = C \ E. For this, let c j (j = 1, 2,... ) be a closed simple curve in Ω E which contains λ 0 = 2, passes through (a j, b j ), and is orthogonal to R. Let c j be oriented clockwise. The fundamental group Γ E is generated by these closed curves. Let J E be the set of all characters on Γ E, that is, the set of all group homomorphisms from Γ E to the unit circle S 1 = R/Z. One puts a group operation on J E via pointwise addition of pairs α 1, α 2 J E : (α 1 + α 2 )(γ) = α 1 (γ) + α 2 (γ) R/Z. Then J E is a compact Abelian topological group. 27

29 Suppose now that E has locally positive measure. We define a particular character δ J E as follows. Let ν be an ergodic measure on Q E, and let w ν be the ν-floquet exponent. Set δ(c j ) = ρ j π R/Z where ρ j = ρ ν (aj,b j ) is the value of the ν- rotation number on (a j, b j ). This character defines a translation flow {ˆτ x } on J E, as follows: ˆτ x (α) = α + δx (α J E, x R). 28

30 Next, introduce an Abel map from the set of divisors D E to the character group J E, as follows. Let ω(λ, F ) be the harmonic measure of a subset F E with respect to the domain Ω E. This means that ω(, F ) solves the Laplace equation ω = 0 in Ω E, and has boundary value equal to the characteristic function of F. Let d = {(y 1, ε 1 ),..., (y j, ε j ),... } D E, and define A(d)(γ k ) = 1 2 bj ε j j=1 a j ω(dλ, E k ) R/Z (6) where E k is the part of E to the right of b k, that is E k = E [b k, ). There is no a priori reason to think that the series on the right-hand side of (6) converges. It does converge if Ω E is a Parreau-Widom domain. 29

31 To explain what this means, we introduce the Green s function G(λ, λ 0 ) of Ω E with logarithmic pole at λ 0 = 2. We assume that Ω E is regular for the Dirichlet problem, which means that G assumes continuously the boundary value zero at each point of E (which is the boundary of Ω E ). Further, let {c j j 1} be the points in Ω E where the gradient G(c j ) = 0; there is exactly one such point in each interval (a j, b j ) and there are no other such points in Ω E. One now says that Ω E is of Parreau-Widom type if j=1 G(c j ) <. 30

32 Theorem (Gesztesy-Yuditskii). Suppose that Ω E is of Parreau-Widom type. Then the Abel map A is well-defined, continuous and surjective, and maps {τ x }-orbits in D E onto {ˆτ x }-orbits in J E. Moreover, there is a right inverse I : J E D E, that is, a map such that A I(α) = α for all α J E. This map I need not be continuous (it is if E is homogeneous in the sense on Carlesen). 31

33 To create sets E and corresponding domains Ω E as in Example 1, we arrange that E satisfies all the hypotheses of the first theorem and is also of Parreau-Widom type. We arrange that the δ-translation flow ˆτ x (α) = α + δx is minimal on J E. We further arrange that the right inverse I : J E D E satisfies the following conditions I is not continuous; I(α + δx) = τ x (I(α)) (α J E, x R) I is of the first Baire class. The last condition means that I(α) = lim I N(α) N (α J E ) where each map I N : J E D E is continuous. This implies that I has a residual set of continuity points. This in turn implies that M = cls{i(α) α J E } D E is an almost automorphic, non-almost periodic minimal subset of D E. 32

34 Example 2 This example will consist of a set E of the the type under discussion for which the divisor map is not injective. Let E = [ 1, ) \ n=1 (a n, b n ), where a = a 0 = 3/4, b = b 0 = 1/2, 0 > b 1 > a 1 > a 2 > b 2 >... and where b j, a j b = 1/2. Note that the interval [ 1, 3/4] E. Let Q E be the corresponding set of potentials. Choose an ergodic measure ν on Q E. (i) One can choose the intervals (a j, b j ) so that b = 1/2 is irregular in the sense of potential theory. This implies that the ν-lyapunov exponent β ν (λ) > 0. (ii) Use the Oseledets Theorem to see that for ν-a.a. q Q E the equation L q ϕ = ( d 2 dx 2 + q(x) ) ϕ = bϕ admits solutions ϕ ± (x) which decay exponentially at x = ±. 33

35 Using the fact that L q ϕ = bϕ is oscillatory, we can find q Q E so that the Dirichlet problem L q ϕ = bϕ ϕ(0) = 0 admits a nonzero solution in L 2 [0, ). (iii) Introduce the spectral measures σ ± of L ± q : σ ± (dλ) = σ ±,ac (dλ) + σ ±,d (dλ) + r ± b λ Choose q Q E so that r + > 0, this is possible by point (ii). If r + 0, r 0, and r + + r = r + + r, then one can use the Marchenko parametrization of GR (and bypass the less informative Gel fand-levitan theory) to show that there exists q Q E with spectral measures σ ± (dλ) = σ ±,ac (dλ) + σ ±,d (dλ) + r ± b λ. Since r + > 0 one has that π is not injective. 34