Chain recurrence, growth rates and ergodic limits

Abstract Averages of functionals along trajectories are studied by evaluating the averages along chains. This yields results for the possible limits and, in particular, for ergodic limits. Applications to Lyapunov exponents and to concepts of rotation numbers of linear Hamiltonian flows and of general linear flows are given.


Introduction
The purpose of this paper is to describe a topological construction by which we obtain characteristics of dynamical systems via ergodic limits obtained by averages along trajectories. We replace limits along trajectories with limits along (ε, T )-chains as ε → 0 and T → ∞. The numbers which are defined by an ergodic limit procedure are then replaced by intervals, each of them corresponding to a maximal chain transitive set, i.e. a chain recurrent component. Therefore, as part of the construction, the chain recurrent components of the relevant flow have to be determined.
In particular, we consider linear flows on vector bundles. For Lyapunov exponents, the relevant flow is then the induced flow on the projective bundle and the construction yields the Morse spectrum as introduced in [4]. For rotation numbers of linear Hamiltonian flows in R 2n (e.g. [7]) the induced flow on the n-dimensional Lagrange subspaces is the relevant one. For rotation numbers in the sense of San Martin [11] (for the stochastic case see also Arnold [1]) the induced flow on the bundle of oriented 2-planes has to be considered. In all cases the corresponding maximal chain transitive sets are described, since they determine the numbers of intervals of Lyapunov exponents and of rotation numbers.
The contents of the paper are as follows. Section 2 presents the construction of limit intervals corresponding to chain transitive sets. Section 3 discusses rotation numbers for linear Hamiltonian flows and §4 discusses rotation numbers of oriented planes for arbitrary linear flows.

Growth rates and chain transitivity
This section describes a general construction for flows on fiber bundles of metric spaces which relates growth rates along trajectories and, in particular, ergodic limits, to evaluations along chains.
In the following a (locally trivial) fiber bundle with typical fiber E is considered which is given by a continuous map π : X → B of metric spaces and a finite open cover (U α ) of B together with homeomorphisms ϕ α : π −1 (U α ) → U α × E, z → (ϕ 1 α (z), ϕ 2 α (z)), satisfying ϕ 1 α (z) = π(z). We will always assume that the base space B is compact. Let be a (continuous) fiber-preserving flow on the fiber bundle π : X → B given by t (x) = (t, x). By local triviality, induces a flow π on the base space which, for simplicity, is denoted by b · t, b ∈ B and t ∈ R. Furthermore, let f : X → R m be continuous. We are interested in the growth rates along trajectories for t → ∞ given by We impose the following two basic assumptions. The map is well defined and the growth rates are uniformly bounded, i.e.
C := sup x∈X lim sup Assumption (2) immediately implies that cluster points for t → ∞ of (1) are independent of the element x in the fiber over π x.
Remark 2.1. For a linear flow on a vector bundle V → with compact metric base space write v = (ω, x) = x . Then the Lyapunov exponent of v is given by the limit as t → ∞ (if it exists) of (1/t) log (t, v) . Clearly, V is not compact, while the projective bundle PV → is. With only depends on t and Pv ∈ PV. Thus condition (2) is satisfied if we consider as the fiber-preserving flow in the fiber bundle V → PV with base space PV. Furthermore, condition (3) is satisfied, since there are constants K , α > 0 with This implies for t large enough and an analogous lower bound is valid.
Chain recurrence, growth rates and ergodic limits 1511 Remark 2.2. The general problem above includes ergodic limits on a compact metric space B. In fact, consider for a flow b → b · t : B → B and a continuous functional g : Define on the fiber bundle R × B → B the fiber-preserving flow t (s, b) = (s − t, b · t) and the functional Then for x = (s, b) Thus this difference is independent of the element (s, b) in the fiber over b ∈ B, as required in (2); furthermore, (3) is also satisfied, since g is bounded on the compact space B.
In order to describe the limiting behavior of (1) as t → ∞, it is helpful to relax this problem. Instead of studying directly the evaluation along trajectories, we study the evaluation along (ε, T )-chains, and then let ε → 0, T → ∞.
For an (ε, T )-chain ζ in B given by n ∈ N, b 0 = b, b 1 , . . . , b n ∈ B, T 0 , . . . , is independent of the choice of x i . An easy consequence of (3) is that, for all (ε, T )-chains with T large enough, Recall that a set M is chain transitive, if for any two points a, b ∈ M and for all ε, T > 0 there is an (ε, T )-chain from a to b. For a compact invariant chain transitive set M ⊂ B define It is our aim to describe the set F(M) and its relation to the asymptotic growth rates (1) as t → ∞.
We note a number of observations. Proof. By local triviality of the fiber bundle, there exists for b ∈ B a local continuous section σ : U → X defined on a neighborhood U of b with σ • π = id U . Thus for t n → t, b n → b one has with x n = σ (b n ) as claimed. 2 The following lemma [5,Lemma B.2.23] gives a uniform upper bound for the time needed to connect any two points in a chain transitive set. For the convenience of the reader we sketch the proof. LEMMA 2.4. Let M be a compact invariant chain transitive set and fix ε, T > 0. Then there existsT (ε, T ) > 0 such that for all x, y ∈ M there is an (ε, T )-chain from x to y with total length at mostT (ε, T ).
Proof. By assumption, one finds for all x, y ∈ M an (ε/2, T )-chain in M from x to y. Using continuous dependence on initial values and compactness, one finds finitely many (ε, T )-chains connecting every x ∈ M with a fixed z ∈ M. One also finds finitely many (modulo their endpoints) (ε, T )-chains connecting z with arbitrary elements y ∈ M. Thus one ends up with finitely many (ε, T )-chains connecting all points in M. The maximum of their total lengths is the desired upper boundT (ε, T ). 2 The growth rates of concatenated chains are a convex combination of the individual growth rates. LEMMA 2.5. Let ξ, ζ be (ε, T )-chains in B of total lengths σ and τ , respectively, such that the initial point of ζ coincides with the final point of ξ . Then for the concatenated chain ζ • ξ one has Proof. Let the chains ξ and ζ be given by π x 0 , . . . , π x k and π y 0 = π x k , . . . , π y n , with times S 0 , . . . , S k−1 and T 0 , . . . , T n−1 , respectively. Thus the total times are σ = k−1 i=0 S i and τ = n−1 i=0 T i and f (ζ • ξ ) The following proposition shows that it is sufficient to consider periodic chains. Proof. Let λ ∈ F(M) and fix ε, T > 0. It suffices to prove that for every δ > 0 there exists a periodic (ε, T )-chain ζ with |λ − f (ζ )| < δ. By Lemma 2.4 there existsT (ε, T ) > 0 such that for all π x, π y ∈ M there is an (ε, T )-chain ξ in M from π x to π y with total time at mostT (ε, T ). For S > T choose an (ε, S)-chain ζ with |λ − f (ζ )| < δ/2 given by, say, π x 0 , . . . , π x m with times S 0 , . . . , S m−1 > S and with total time σ = m−1 i=0 S i . Concatenate this with an (ε, T )-chain ξ from π x n to π x 0 with points π y 0 = π x n , . . . , π y m = π x 0 , with times T 0 , . . . , T m−1 > T and total time τ = m−1 i=0 T i ≤T (ε, T ). The periodic (ε, T )-chain ζ = ξ • ζ has the desired approximation property: since the chain ξ depends on ζ , τ also depends on ζ . However, the total length of ξ is bounded as By (6) there is a uniform bound for | f (ξ )| and | f (ζ )| for all considered chains ξ and ζ . Since τ remains bounded for chains ζ with total length σ tending to ∞, the right-hand side tends to 0 as S → ∞. 2 The next theorem describes the set of asymptotic growth rates; part (ii) shows that the cluster points for chains in the chain recurrent set comprise all asymptotic growth rates for arbitrary initial points. Together with Proposition 2.6 it is our main result on growth rates.
if the limit exists; here M is the maximal chain transitive set containing ω(π x) ⊂ B.
Proof. (i) The proof is based on a 'mixing' of growth rates. It is clear that F(M) is closed; it is also bounded by (6), and hence compact. Thus it suffices to show that for all λ ∈ coF(M), all δ > 0 and all ε, Denote the initial (and final) point of ζ i by π x i . By chain transitivity there are (ε, T )chains ξ i from π x i to π x i+1 and ξ N from π x n to π x 0 . For k ∈ N let ζ i,k be the k-fold concatenation of ζ i . Then for k 1 , . . . , k N ∈ N the concatenation ζ (k 1 , . . . ,
(ii) Here we start from an arbitrary initial point x ∈ X and have to show that the corresponding limit points can be approximated by chains in ω(π x). Recall that ω-limit sets are connected and contained in the chain recurrent set. Hence the maximal chain transitive set M ⊂ B containing ω(π x) is well defined for x ∈ X and the inclusion is obvious. Thus it suffices to show the following.
Let λ(x) be a cluster point of (1/t) f ( (t, x)) for t → ∞. For all δ > 0 and all Fix δ > 0, ε > 0 and T > 1. By uniform continuity of π on the compact set Invoking Lemma 2.3 one can also require that holds for all t ∈ [0, 2T ]. Furthermore, one may assume that By definition there are t n → ∞ with where one can assume without loss of generality that π x · t n converges for t n → ∞. Now fix N ∈ N such that for all n ≥ N d(π x · t n , π x · t N ) < ε Setting σ n = t n − t N we have Choose n large enough such that with T 0 = σ n we have T 0 > 2T and Clearly, (12) (13) and (14), we obtain in addition to (12) and We partition the interval [0, T 0 ] into pieces of length τ j with T ≤ τ j < 2T for j = 0, . . . , l − 1. Thus Set y 0 = x and y j+1 = (τ j , y j ) for j = 0, . . . , l − 1. Then (T 0 , x) = y l and Define an (ε/3, T )-chainζ in X by l ∈ N, τ 0 , . . . , τ l−1 ≥ T, π y 0 , . . . , π y l−1 , π y 0 ∈ X , noting that from (15) we have Using (18) we obtain However, the chainζ is not necessarily contained in ω(π x). In order to obtain an appropriate chain ζ in ω(π x), we use (10) and (12).

F. Colonius et al
The special case of the average functional (4) has the advantage that one can also use methods from ergodic theory for its analysis leading to the following results.

Rotation numbers for linear Hamiltonian systems
In this section, we will apply the general results above to linear Hamiltonian flows and rotation numbers. For the definition of the rotation number, we follow the exposition in Fabbri et al [7, §2] without, however, specifying an ergodic measure on the base.
Consider a family of linear Hamiltonian systems Here where I n is the identity matrix, is a compact metric space with continuous flow σ abbreviated by σ (ξ, t) = ξ · t and H is a continuous 2n × 2n matrix-valued function on Chain recurrence, growth rates and ergodic limits 1517 with values H (ξ ) in the real symmetric 2n × 2n matrices (equivalently, J −1 H ∈ sp(n, R), the algebra of infinitesimally symplectic matrices). The solutions induce a skew product flow on × L R where L R is the space of all real n-dimensional Lagrangian planes of R 2n . This space is a compact manifold of dimension n(n + 1)/2. Let U (t, ω) be the fundamental solution of (21). Then U (t, ω)l 0 ∈ L R for l 0 ∈ L R . Hence we obtain a linear skew product flow on K R = × L R given by Recall that the space L R of Lagrange planes can be identified with U (n)/O(n). Following Arnold [2] define Note that this is independent of the representation by u, since u · O(n) = u · O(n) implies u = u O for an orthogonal matrix O, and hence Finally, for an ergodic measure µ on , define the rotation number α by where arg is any argument of a complex number. Then µ-almost everywhere this limit exists and is constant. (Note that the argument here is independent of the branch, but follows one of them; thus we may choose arg Det 2 l 0 as the principal value in [0, 2π ).) Furthermore, the limit in (23) can also be written in the form (4) as an ergodic limit. The limit in (23) fits into the framework of §2. Define the continuous function f : X = × L R → R, f (ω, l 0 ) = arg Det 2 l 0 .
Our previous results yield the following. For every ergodic invariant measure ν on K R the corresponding rotation number satisfies where M is the maximal chain transitive set containing the support of ν. Furthermore, the boundary points α * (M) and α(M) are attained for certain ergodic invariant measures on K R . Remark 3.2. In order to analyze rotation numbers corresponding to ergodic measures µ on the base space , we have to lift them to ergodic measures ν on × L R projecting down to µ. This is always possible; however, there may be several possibilities for ν (depending, in particular, on the choice of the maximal chain transitive set containing the support of ν).
The next step is the classification of the maximal chain transitive sets in K R . If the flow on is chain transitive and locally transitive this has been done by Braga Barros and San Martin [3], who show in particular that their number is bounded by 2 n . Their setting is as follows.
The space L R of n-dimensional Lagrangian subspaces of R 2n is a compact manifold embedded in the Grassmannian Gr n (2n) of n-dimensional subspaces of R 2n . The group Sp(n, R) acts transitively on L R . We cite the following definition. Under the assumption of local transitivity the chain recurrent components of can be described by fixed point sets (in L R ) of diagonalizable matrices in Sp(n, R). Consider a diagonal matrix of the form where = diag{λ 1 , . . . , λ n }, a n × n diagonal matrix with λ i > 0, i = 1, . . . , n. Such a matrix belongs to Sp(n, R). Put for the adjoint orbit of D. To have this orbit we can choose = diag{λ 1 , . . . , λ n } so that λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ 1 (since there are matrices g ∈ Sp(n, R) such that the conjugation g Dg −1 just permutes the entries of ). The chain recurrent components of are described in terms of O( ) as follows [3]. THEOREM 3.4. Assume that the base flow on is chain transitive and that the local group of local homeomorphisms on is locally transitive. Then there exist ∈ Sp(n, R) and a map D : → O( ) such that any chain recurrent component has the form The fixed point set which enters in each union is taken in a compatible way. The following example in the regular situation clarifies the meaning of compatible. Suppose that, for a given flow , = diag{λ 1 , . . . , λ n } is such that λ 1 > · · · > λ n ≥ 1. Denote by {e 1 , . . . , e n , f 1 , . . . , f n } the basis which diagonalizes D (ω). There are 2 n fixed points of D (ω) in L R , namely the n-dimensional Lagrangian subspaces spanned by basic elements (that is, span{e i(1) , . . . , e i(k) } ∪ { f j (1) , . . . , f j (n−k) } with {i 1 , . . . , i k } ∩ { j 1 , . . . , j n−k } = ∅). Thus there are 2 n chain transitive components of the flow in L R . To see how they are built above the base space note that has just one attractor, namely att( ) = span{e 1 , . . . , e n }. In the same way, D (ω) also has one attractor att(D (ω)). This way, ω∈ att(D (ω)) is a chain transitive component, which turns out to be an attractor of . Also, (and each D (ω)) has just one repeller (span{ f 1 , . . . , f n }) fixed point and these repellers are combined together to give a repeller component of .
Analogously, the other fixed points of D (ω) ∈ O( ), ω ∈ , can be labelled (in terms of bases like {e 1 , . . . , e n , f 1 , . . . , f n }) in a consistent way to give all the 2 n components.
In the case when is not regular (that is, there is repetition of eigenvalues) the picture is similar. The main difference is that a repetition of the eigenvalues forces a collapsing of the components.
This implies the following result.
COROLLARY 3.5. Consider a non-autonomous linear Hamiltonian differential system (21) and the associated linear Hamiltonian skew product flow K R given by (22). Assume that the base flow on is chain transitive and that the local group of local homeomorphisms on is locally transitive. Then there are at most 2 n compact intervals of rotation numbers, each of them corresponding to a chain recurrent component M of the flow on K R .
Since the rotation number is specified by the behavior of a Lagrange plane, the corresponding Lyapunov exponent also has to be specified for Lagrange planes. Thus, one has to look at the growth rates in the corresponding exterior product space. For the Morse spectrum this was analyzed in [6]. One can either study these Lyapunov exponents separately or, and this appears to be more adequate, study the exponential growth rates (of the Lagrange planes) and the rotation numbers simultaneously. Thus we define the continuous map Conditions (2) and (3) are satisfied, and we obtain the following result.
COROLLARY 3.6. Consider a non-autonomous linear Hamiltonian differential system of the form (21) and the associated linear Hamiltonian skew product flow on K R given by (22). Assume that the base flow on is chain transitive and that the local group of local homeomorphisms on is locally transitive. Then, for each of the (at most) 2 n chain recurrent components M i of the flow on K R , the set Remark 3.7. It may appear more natural to consider f in (24) as a map into the complex numbers C where the Lyapunov exponent is the real part and the rotation number is the imaginary part. Thus F(M i ) is a compact convex subset of C.
Remark 3.8. Braga Barros and San Martin [3] show that local transitivity holds, e.g. if X is a compact Riemannian manifold. It is not clear if local transitivity holds for the closure of an almost periodic function or for the set U = {u ∈ L ∞ (R, R m ), u(t) ∈ U for almost all t ∈ R} of functions with values in a compact and convex set U ⊂ R m , endowed with the weak * topology on L ∞ . The latter space is of relevance in control theory (cf. [5]).
Remark 3.9. If we omit the assumption of local transitivity, we conjecture that the methods of Salamon and Zehnder [10] combined with [6] (concerning general linear flows on vector bundles) can be modified so that they yield the existence of the finest Morse decomposition for the flow on Lagrange planes. It would therefore follow that the number of maximal chain transitive sets is finite.
where ρ ≥ 0 is a parameter and U is a compact and convex subset of R m . We can then take as the base flow the shift on the space U of perturbation (or control) functions. One will expect semicontinuity results of rotation numbers and Lyapunov exponents depend on ρ. Under an inner-pair condition (cf. Gayer [8] for methods of verification) continuous dependence on ρ may also be expected.

Rotation numbers in planes
An alternative generalization of rotation numbers has been proposed by San Martin [11]. It relies on the restriction of the flow to oriented planes. We follow the presentation of Arnold [1].
Recall that the free operation G × E → E of a finite group G on a Hausdorff space E defines a principal G-bundle (e.g. tom Dieck [12, Example 6.5]) For rotation numbers the Grassmann manifold G + 2 (d) of oriented 2-planes has to be considered. It is a twofold covering of the Grassmann manifold G 2 (d). It is helpful to also consider the Stiefel manifold St 2 (d) of orthonormal 2-frames which is a principal bundle over G 2 (d) with structure group S O(2, R). One also has the principal fiber bundle over the (1, 2)-flags with structure group G ∼ = Z 2 2 , the Kleinian group of four elements. More explicitly, for a flag (V 1 ⊂ V 2 ) ∈ F (1,2) an element n = (u, v) ∈ St 2 (d) is given by u ∈ V 1 with u = 1 and v ∈ V 2 with v = 1 and u, v = 0. These two vectors give rise to four orthonormal frames, namely n 1 = n = (u, v), n 2 = (−u, v), n 3 = (−u, −v) and n 4 = (u, −v). Here n 1 and n 3 have the same orientation while n 2 and n 4 have the same, but opposite, orientation. The action of G = {g 1 , g 2 , g 3 , g 4 } is defined by g i n = n i , i = 1, . . . , 4. Thus (25) is a fourfold covering.
This generalizes to vector bundles Remark 4.1. Analogous definitions can be given for the Stiefel manifold over a Riemannian manifold (cf. Arnold [1]). Note that here the rotation number (defined below) depends on the Riemannian metric.
Let be a smooth linear flow on V. This induces smooth flows on the oriented 2-planes G + 2 (V) and on the flag bundle F (1,2) (V). It also induces a smooth flow on the Stiefel bundle which is defined as follows. Take a frame n = (u, v) and ω ∈ . Then for t ∈ R define the image at time t as the orthonormalized pair This defines a flow on St 2 (V) which, as for those on the Grassmann bundles and on the flag bundle, we also denote for simplicity by . Then is a fiber bundle with structure group G = Z 2 2 and compact base space. The induced flows are equivariant with respect to G.
The next proposition relates the maximal chain transitive sets in the bundle and in the base. Then for δ > 0 small enough, a δ-chain in M B from b to b gives rise to ε-chains from each of the e i to some e j (i) . Since π −1 (M B ) is compact invariant, it contains a maximal chain transitive set which is then invariant. Hence there is a maximal chain transitive set M E in π −1 (M B ) which projects to M B . By uniform continuity, a chain transitive set in E projects down to a chain transitive set in B. Thus, by maximality of M B , the set M E is also a maximal chain transitive set in E (not just in π −1 (M B )). For every g ∈ G the set g M E is also a maximal chain transitive set.
Every maximal chain transitive set maps down into a maximal chain transitive set in B. The arguments above show that it is of the form above and hence the projection is onto; thus the assertion follows. 2 As a consequence of this result one can give estimates on the number of maximal chain transitive sets in Stiefel bundles and in oriented 2-plane bundles. where ω · t denotes the induced flow on the base. For a chain ζ in St 2 (V) given by (u 0 , v 0 , ω 0 ), . . . , (u n , v n , ω n ) ∈ St 2 (V), T 0 , . . . , T n > T , define the rotation number of ζ as Recall that the chain recurrent components in St 2 (V) project down to the chain recurrent components in G + 2 (V).  Then it follows (cf. [1]) that the limit does not change if we start in another orthonormal frame in the same fiber over G + 2 (V). Thus the boundary point ρ * is also attained for each other chain recurrent component in St 2 (V) projecting down to M. 2