GENERALIZED EXPONENTIAL BEHAVIOR AND TOPOLOGICAL EQUIVALENCE

. We discuss the topological equivalence between evolution families with a generalized exponential dichotomy. These can occur for example when all Lyapunov exponents are inﬁnite or all Lyapunov exponents are zero. In particular, we show that any evolution family admitting a generalized exponential dichotomy is topologically equivalent to a certain normal form, in the which the exponential behavior in the stable and unstable directions are multiples of the identity. Moreover, we show that the topological equivalence between two evolution families admitting generalized exponential dichotomies, possibly with diﬀerent growth rates, can be completely characterized in terms of a new notion of equivalence between these rates.


1.
Introduction. In the theories of differential equations and dynamical systems, the emphasis is often on the description of the qualitative behavior of the system. Sometimes this is unavoidable, due to the lack of an explicit description of the trajectories, although also often it is simply the best approach to understand some properties of the system. For example, the Grobman-Hartman theorem says that in a sufficiently small neighborhood of a hyperbolic fixed point there is a topological conjugacy between the trajectories of the system and those of its linearization. Hence, at least in a neighborhood of the hyperbolic fixed point, from the qualitative point of view it is sufficient to know the associated linear system. Certainly, there are other types of conjugacies, such as differentiable conjugacies, and they play an important role in the theory although in our setting they would be too rigid. For example, two linear flows in a finite-dimensional space are differentially conjugate if and only if the matrices of the systems are conjugate, that is, if they have the same Jordan canonical form.
In this work our main aim is to discuss the topological equivalence between (linear) evolution families with a generalized exponential dichotomy, in the sense that the contraction and expansion can be of the form e cρ(t) for an arbitrary increasing function ρ, instead of simply the usual exponential behavior e ct . This type of generalized exponential behavior was considered earlier for example in [1,4,5]. It can occur naturally for instance when all Lyapunov exponents are infinite or all Lyapunov exponents are zero (the latter may correspond, for example, to a polynomial behavior).
Before being more detailed, we first recall briefly how evolution families appear naturally. Consider a nonautonomous linear equation for all t, s, r. Any such family of linear operators is a particular case of an evolution family (see Section 2 for the definition). In the particular case considered here the map (t, s) → U (t, s) is differentiable but we shall not require this property. For example, we can also consider evolution families that may be obtained from nonautonomous linear equations with unbounded linear operators A(t) (we refer the reader to the book [7] for details on the basic theory).
In order to describe our main results we recall briefly the notions of an exponential dichotomy and of a topological conjugacy. The classical notion of an exponential dichotomy corresponds to the existence of complementary projections P (t) and Q(t) for each t and constants K, λ > 0 such that for t ≥ s. In the notion of a generalized exponential dichotomy these inequalities are replaced by for t ≥ s, thus allowing arbitrary growth rates. We also recall that two evolution families U (t, s) and V (t, s) are said to be topologically equivalent if there exist homeomorphisms h t (sometimes with a prescribed growth at infinity; see Section 3) such that for all t, s. The maps h t act as a dictionary between the two dynamics and any properties depending only on the topology, such as for example the existence of dense orbits, is transferred by these maps from one dynamics to the other. Our main results are the following: 1. Any evolution family admitting a generalized exponential dichotomy is topologically equivalent to a certain normal form, in the which the behaviors in the stable and unstable directions are multiples of the identity and so they are essentially 1-dimensional. 2. The topological equivalence between two evolution families admitting generalized exponential dichotomies, possibly with different growth rates, can be completely characterized by a new notion of equivalence between growth rates.

GENERALIZED EXPONENTIAL BEHAVIOR AND TOPOLOGICAL EQUIVALENCE 3025
Our first main result says that an evolution family U (t, s) admitting a generalized exponential dichotomy can be brought by a topological conjugacy to the normal form where x and y belong, respectively, to spaces whose dimensions are those of the stable and unstable subspaces. The proof consists of constructing explicitly the maps h t in the notion of topological equivalence. For a nonautonomous linear differential equation on an arbitrary Banach space, it was shown in [8] that if the equation admits an exponential dichotomy (with ρ(t) = t), then the corresponding evolution family is topologically conjugate to the evolution family associated to (1) (the case of finite-dimensional spaces was considered earlier in [6]). Theorem 3.1 further generalizes this result to generalized exponential dichotomies, with a proof based on an explicit construction of conjugacies inspired in [8].
A consequence of Theorem 3.1 is that any evolution family U (t, s) with generalized bounded growth (see Section 2 for the definition) that admits a generalized exponential dichotomy is topologically equivalent to any sufficiently small linear perturbation where t → B(t) is a continuous map with values on the set of bounded linear operators on E. Indeed, it can be shown that if sup ξ∈R B(ξ) is sufficiently small, then the evolution family V (t, s) also admits a generalized exponential dichotomy (with the same growth rate) and with topologically equivalent stable and unstable subspaces (see [2]). Therefore, as a consequence of Theorem 3.1, the evolution families U (t, s) and V (t, s) are topologically equivalent. In the particular case of the growth rate ρ(t) = t, this result was obtained earlier in [8].
Our second main result addresses the problem of how one can characterize the notion of topological equivalence between evolution families admitting a generalized exponential dichotomy in terms of a notion of equivalence between growth rates. Two growth rates ρ 1 and ρ 2 are said to be equivalent if there exist constants α i , β i > 0 for i = 1, 2 such that for t ≥ s. It turns out that this notion characterizes completely (and is in fact equivalent to) the notion of topological equivalence between evolution families admitting generalized exponential dichotomies, possibly with different growth rates ρ 1 and ρ 2 . This is the content of Theorem 4.4. As an outcome one can classify completely all 1-dimensional evolution families U (t, s) = e ρ(t)−ρ(s) using the notion of equivalence (see Proposition 4.6). Finally, in Section 5 we detail briefly the relations between Bohl exponents and the notion of equivalence between growth rates. 1. U (t, t) = Id and U (t, s)U (s, r) = U (t, r) (2) for t ≥ s ≥ r; 2. for each s ∈ R and x ∈ E, the map t → U (t, s)x is continuous. An evolution family U is said to be reversible if U (t, s) is invertible for all t ≥ s. In this case we write U (s, t) = U (t, s) −1 for t ≥ s and identity (2) holds for all t, s, r ∈ R. For simplicity of the exposition we shall always consider reversible evolution families. Now let F be the set of all increasing continuous functions ρ : R → R with ρ(0) = 0. Given ρ ∈ F, we write A reversible evolution family U is said to have ρ-bounded growth if there exist constants M, α ≥ 0 such that The following is a characterization of bounded growth.
Proposition 2.1. A reversible evolution family U has ρ-bounded growth if and only if there exist constants ε, δ > 0 such that for each t, s ∈ R.
Proof. It is clear that if U has ρ-bounded growth, then (3) holds. Now let us assume that property (3) holds. Given t ≥ s, let where [·] denotes the integer part. Then

By (3) we obtain
where c = log ε/δ. One can proceed in a similar manner for t ≤ s and hence, U has ρ-bounded growth.
3. Exponential dichotomies and topological equivalence. In this section we consider evolution families admitting a generalized exponential behavior and we show that they can always be transformed (using a topological conjugacy) into a canonical form that expands and/or contracts the same in all directions. A reversible evolution family U is said to admit an ρ-exponential dichotomy if there exist complementary projections P (t) + Q(t) = Id, for t ∈ R, and constants K, λ > 0 such that We also introduce the notion of topological equivalence. Two reversible evolution families U = {U (t, s)} and V = {V (t, s)} are said to be topologically equivalent if there exist a continuous map h : R × E → E and an increasing onto map L : Our main result shows that any reversible evolution family admitting an exponential dichotomy can be brought to an essentially one-dimensional evolution family by an appropriate topological conjugacy.
Theorem 3.1. Given a C 1 onto function ρ ∈ F, let U be a reversible evolution family with ρ-bounded growth. If U admits a ρ-exponential dichotomy, then it is topologically equivalent to the evolution family Proof of the lemma. Clearly, where We have and similarly, This yields the second inequality in (4). Moreover, for ξ ≥ t we have for some constants M, α > 0 (since U has ρ-bounded growth). Therefore, and analogously, The first inequality in (4) follows now readily from (5)- (6) and (7)- (8).
The following properties hold: Proof of the lemma. Since we have d dt Similarly, since This establishes the first property. Now we establish the second property. It follows from (9) that U (t)P x t → 0 when t → ∞. On the other hand, for t ≤ ξ ≤ 0 we have and thus, Therefore, U (t)P x t = ∞ when t → −∞, provided that P x = 0. One can show in a similar manner that provided that Qx = 0. The desired property follows now readily from the first property.
We continue with the proof of the theorem. Define functions

GENERALIZED EXPONENTIAL BEHAVIOR AND TOPOLOGICAL EQUIVALENCE 3029
respectively, by and . In order to show that the maps h t give the desired topological conjugacy, we divide the proof into several steps.
Step 1. Invariance. We first show that We have and analogously, where V (t) = V (t, 0). This readily implies that Step 2. Injectivity of the maps h t . Assume that h P t (P x) = h P t (P y). where and τ 2 = ρ −1 (ρ(t) + log P y ).
The injectivity of the maps h Q t can be proved in a similar manner. This readily implies that the maps h t are one-to-one.
Step 3. Surjectivity of the maps h t . Take P (t)y ∈ P (t)E. If h P t (P x) = P (t)y, then (14) Therefore U (τ )P U −1 (t)y τ = 1 (the existence and uniqueness of τ is guaranteed by Lemma 3.3). By (14), we obtain Moreover, for a unique η (its existence and uniqueness is guaranteed by Lemma 3.3). This shows that h P t and h Q t are invertible, with inverses and The inverse h −1 t : E → E is now given by Step 4. Existence of the map L. By (4) and (10), for If P x ≤ 1, then since e Φ(τ,t) = P x we have τ ≤ t and

GENERALIZED EXPONENTIAL BEHAVIOR AND TOPOLOGICAL EQUIVALENCE 3031
If P x > 1, then τ > t and Therefore, Similarly, by (11), we obtain On the other hand, if Qx > 1, then t > η and Therefore, (17). Now we observe that by (15), where τ is determined by the identity

LUIS BARREIRA, LIVIU HORIA POPESCU AND CLAUDIA VALLS
If t ≥ τ , then Using (6) we obtain (without loss of generality one can always assume that M λ > Kα). On the other hand, if t < τ , then Therefore, By (19) and (20), we obtain Similarly, by (16), we have where η is determined by the identity One can show in an analogous manner that if t < η, then The above estimates yield the desired result. Indeed, . This concludes the proof of the theorem. 4. Equivalence relations. In this section we consider a notion of equivalence between the growth rates in F and we show that it characterizes completely the notion of topological equivalence between evolution families (not necessarily admitting an exponential dichotomy).
Now we define a binary relation on the set F by for some constant α > 0. Equivalently, The following is a characterization of the binary relation in (22).
On the set F we define an equivalence relation that is related to the notion of topological equivalence. Two functions ρ 1 , ρ 2 ∈ F are said to be equivalent and we write ρ 1 ∼ ρ 2 if ρ 1 ρ 2 and ρ 2 ρ 1 . For example, the functions ρ 1 (t) = t 3 and ρ 2 (t) = t 3 + t are equivalent. Using Proposition 4.2 we readily obtain a characterization of the notion of equivalence.

4.3.
Exponential dichotomies and equivalence. Now we relate in an optimal manner the notions of equivalence and of topological equivalence for evolution families that admit exponential dichotomies.
Theorem 4.4. Given onto functions ρ 1 , ρ 2 ∈ F, let U i be a reversible evolution family with ρ i -bounded growth, for i = 1, 2. If U i admits a ρ i -exponential dichotomy with projections P i +Q i = Id, for i = 1, 2, then U 1 and U 2 are topologically equivalent if and only if: 1. the subspaces P 1 E and P 2 E are homeomorphic; 2. the subspaces Q 1 E and Q 2 E are homeomorphic; 3. the functions ρ 1 and ρ 2 are equivalent.
Proof. We start with an auxiliary result. Given ρ 1 , ρ 2 ∈ F, consider the evolution families U 1 and U 2 formed by the linear operators We shall write U 1 ∼ U 2 to mean that U 1 and U 2 are topologically equivalent. Proof of the lemma. We first assume that the evolution families are topological equivalent. It follows from Now take ε, δ > 0 such that e δ = L(e ε L(1)). If Φ 1 (t, s) < ε for some t ≥ s, then Φ 2 (t, s) < δ. By Proposition 4.2, we conclude that ρ 2 ρ 1 . One can show in a similar manner that ρ 1 ρ 2 and hence, ρ 1 and ρ 2 are equivalent. Now we assume that ρ 1 and ρ 2 are equivalent. By Proposition 4.3, there exist functions ϕ, ψ : R → R satisfying (27) and (28) for t, s ∈ R. We define ε : R → R by for each t ∈ R. One can easily verify that h t is invertible, with inverse The continuity of the functions h t and h −1 t at 0 follows directly from It remains to show that there exists a map L as in the notion of topological equivalence. For x = 0 we have . Therefore, x = e Φ1(t,s) and using inequality (28) we obtain h t (x) = e Φ2(t,s) ≤ e ψ(Φ1(t,s)) = e ψ(log x ) = L 1 ( x ), where L 1 (θ) = e ψ(log θ) , θ > 0, 0, θ = 0.
We proceed with the proof of the theorem. Let us consider the evolution families V 1 , V 2 , V 2 formed by the linear operators V i (t, s) = e Φi(s,t) P i + e Φi(t,s) Q i , i = 1, 2, and V 2 (t, s) = e Φ1(s,t) P 2 + e Φ1(t,s) Q 2 . By Theorem 3.1, we have U i ∼ V i for i = 1, 2.
We first assume that properties 1-3 hold. Since the subspaces P 1 E and P 2 E are homeomorphic, their unit spheres S(P 1 E) and S(P 2 E) are also homeomorphic. The same happens to the spheres S(Q 1 E) and S(Q 2 E). Let f : S(P 1 E) → S(P 2 E) and g : S(Q 1 E) → S(Q 2 E) be homeomorphisms. We define maps F : P 1 E → P 2 E and G : One can easily verify that F and G are homeomorphisms, with inverses We have and hence, V 1 ∼ V 2 . Since ρ 1 and ρ 2 are equivalent, by Lemma 4.5 we have V 2 ∼ V 2 . Moreover, since V 1 ∼ V 2 , we obtain V 1 ∼ V 2 and since U i ∼ V i for i = 1, 2 (by Theorem 3.1), we conclude that U 1 ∼ U 2 . Now we assume that U 1 ∼ U 2 . Since U i ∼ V i for i = 1, 2, we obtain V 1 ∼ V 2 . Hence, there exists a family of homeomorphisms h t : E → E such that h t e −ρ1(t) P 1 x + e ρ1(t) Q 1 x = e −ρ2(t) P 2 h 0 (x) + e ρ2(t) Q 2 h 0 (x).
But this happens if and only if Q 2 h 0 (P 1 x) = 0 or, equivalently, Therefore, h 0 (P 1 E) ⊂ P 2 E.
Relations (35) and (36) imply that the subspaces P 1 E and P 2 E are homeomorphic, via h 0 . One can show in an analogous manner that the subspaces Q 1 E and Q 2 E are homeomorphic. Proceeding as above, one can now show that V 1 ∼ V 2 and hence V 2 ∼ V 2 . Thus, there exists a family of homeomorphisms g t : E → E such that g t e Φ1(s,t) P 2 x + e Φ1(t,s) Q 2 x = e Φ2(s,t) P 2 + e Φ2(t,s) Q 2 g s (x).
As above, one can show that the subspaces P 2 E and Q 2 E are homeomorphic. Moreover, g t (e Φ1(s,t) P 2 x) = e Φ2(s,t) g s (P 2 x) and g t (e Φ1(s,t) Q 2 x) = e Φ2(s,t) g s (Q 2 x).
Applying Lemma 4.5 to the subspaces P 2 E and Q 2 E, we conclude that the functions ρ 1 and ρ 2 are equivalent.
As an outcome of our approach one can classify completely all 1-dimensional evolution families U (t, s) = e ρ(t)−ρ(s) . The following is an immediate consequence of Proposition 4.3 and Lemma 4.5.

Generalized Bohl exponents.
In this section we detail the relations between Bohl exponents and the notion of equivalence between growth rates. We refer the reader to [3] for the classical notion of Bohl exponent.
Let U be an evolution family. The upper and lower ρ-Bohl exponents of U are defined, respectively, by More precisely, B(ρ, U) is the infimum of all numbers γ ∈ R such that log U (t, s) Φ(t, s) < γ for t, s ∈ R, with min{t − s, s} sufficiently large. The number B(ρ, U) is defined in a similar manner. Clearly, U has ρ-bounded growth whenever B(ρ, U) < ∞ and B(ρ, U) > −∞.
The following result establishes some basic relations between the Bohl exponents of equivalent functions.