Topological Conjugacy for Lipschitz Perturbations of Non-Autonomous Systems

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2016017 DYNAMICAL SYSTEMS Volume 36, Number 9, September 2016 pp. 5011–5024

TOPOLOGICAL CONJUGACY FOR LIPSCHITZ PERTURBATIONS OF NON-AUTONOMOUS SYSTEMS

Ming-Chia Li∗ and Ming-Jiea Lyu Department of Applied Mathematics National Chiao Tung University, 1001 University Road Hsinchu 300, Taiwan

(Communicated by Lan Wen)

Abstract. In this paper, topological conjugacy for two-sided non-hyperbolic and non-autonomous discrete dynamical systems is studied. It is shown that if the system has covering relations with weak Lyapunov condition determined by a transition matrix, there exists a sequence of compact invariant sets restricted to which the system is topologically conjugate to the two-sided subshift of ﬁnite type induced by the transition matrix. Moreover, if the systems have covering relations with exponential dichotomy and small Lipschitz perturbations, then there is a constructive veriﬁcation proof of the weak Lyapunov condition, and so topological dynamics of these systems are fully understood by symbolic representations. In addition, the tolerance of Lipschitz perturbation can be characterised by the dichotomy tuple . Here, the weak Lyapunov condition is adapted from [12, 24, 15] and the exponential dichotomy is from [2].

1. Introduction. In the study of dynamical systems, getting topological conjugacy to a symbolic system is a popular method to realize the long-term qualitative behavior of a system from topological viewpoints. Under uniformly hyperbolic condition, the classical results are presented by Smale [21] for horseshoe maps, Hirsch and Pugh [8] for hyperbolic maps, and Robinson [19] for structurally stable maps, etc. Without hyperbolicity, there are computable methods to obtain topological semi-conjugacy; for instance, Zgliczy´nski[22, 23] introduced covering relations, Mis- chaikov et al. [4,5, 13] used discrete Conley index, Yorke et al. [10,9] presented expanders family, and Richeson et al. [7, 17] for index systems. Furthermore to obtain topological conjugacy, in our previous work [15, 16], we brought in the Lya- punov condition, which is adapted from the cone condition in [12, 24]. In this paper, we weaken the Lyapunov condition and establish a method to verify it, but still secure the truth of the conjugacy result. Non-autonomous discrete systems are time-dependent so that the maps describ- ing the evolution are allowed to change with time; while autonomous systems are time-independent and can be viewed as a particular case of non-autonomous ones.

2010 Mathematics Subject Classiﬁcation. 37B55, 37D25, 37C15, 37B10. Key words and phrases. Topological conjugacy, exponential dichotomy, Lipschitz perturbation, non-autonomous systems, nonuniformly hyperbolic systems. This work was partially supported by MOST grant 104-2115-M-009-003-MY2. The authors are grateful to the anonymous referee for valuable suggestions. ∗ Corresponding author: M.-C. Li.

5011 5012 MING-CHIA LI AND MING-JIEA LYU

Such systems arise quite naturally in practical applications, for example, the Eu- ler method with a variable time-step hn > 0 for a differential equationx ˙ = f(x) gives a non-autonomous system of the form xn+1 = xn + hnf(xn). In recent years there has been growing interest in non-autonomous problems; refer to [1,6] for surveys with open problems and [11, 18] for monographs in this topics. For nonuniformly hyperbolic dynamics, Barreira and Valls [2] established a version of the Grobman-Hartman theorem by constructing topological conjugacy between a linear non-autonomous system with non-uniform exponential dichotomy and its small Lipschitz perturbations. Moreover, in [3], they discussed the relation between non- uniform exponential dichotomy and strict Lyapunov sequence. For motivational purpose, let us illustrate interesting dynamics with the following simple example. For each integer n, let Dn be a constant such that 1 ≤ Dn ≤ 3 if n ≥ 0 and 2 ≤ Dn ≤ 3 if n < 0, and define a map fn on the real line by ( Dnx − 1 + Ln, for x ∈ R and Dn ≥ 2, fn(x) = 2 2 (1) ±Dnx ± Ln x + sin (2πx) , for x Q 0 and Dn < 2, where Ln > 0 is a small constant. Then (fn)n∈Z forms a two-sided non-autonomous system with linear part (Dn)n∈Z and Lipschitz perturbation involving (Ln)n∈Z.A bounded orbit of the system is a two-sided bounded sequence (xn)n∈Z of numbers satisfying xn+1 = fn(xn) for all integers n. It is clear that if Dn ≥ 2 for all n, then P∞ Qi −1 xn ≡ −1 + i=n(Di − Li) j=n Dj is the unique bounded orbit of system (1) and it persists under small Lipschitz perturbations. If Dn = 1 for all n ≥ 0, then Pn Qi −1 xn ≡ i=−1(1 − Li) j=n Dj for n < 0 and xn ≡ 0 for n ≥ 0 form a bounded orbit for the system; nevertheless, the uniqueness of bounded orbits requires the assumption that the one-sided sequence (Ln)n≥0 is not summable, and hence it dose not necessarily persist under perturbations. For topological conjugacy from system (1) to its small perturbations, it gives rise to the question whether a unique bounded orbit persists under perturbations. It is not difficult to see that the answer is yes if the limit inferior of Dn is not equal to one as n goes to infinity, while the system is uniformly hyperbolic in some sense. However, the answer is no if the Q∞ limit of Dn is equal to one as n goes to infinity, the infinite product n=0 Dn of the one-sided sequence (Dn)n≥0 is convergent, and the one-sided sequence (Ln)n≥0 is summable, while the system is neither uniformly hyperbolic nor exponentially dichotomic. This leads us to consider the general case when the system is not uniformly hyperbolic but is exponentially dichotomic. Notice that if one of Dn’s is less than 2, then we cannot apply the result of [15, 16], since a Lyapunov condition of quadratic form is not satisfied. In this paper, instead of just a unique bounded orbit in the above example, we consider the general case, namely, bounded invariants sets for high-dimensional systems, possibly with both stable and unstable directions. We first use covering relations of Zgliczyński[22, 23] to change local coordinate along stable and unstable directions, then impose weak Lyapunov condition in form of a sequence of real- valued functions to track displacements of two orbits, and finally obtain topological conjugacy. More precisely, for a two-sided non-autonomous system having covering relations with weak Lyapunov condition determined by a transition matrix A, we show that the system has a sequence of compact invariant sets restricted to which the system is topologically-conjugate to σA, the two-sided subshift of finite type induced by A. TOPOLOGICAL CONJUGACY FOR NON-AUTONOMOUS SYSTEMS 5013

In addition, we show that if the system has linear part admitting exponential dichotomy of Barreira and Valls [2] and non-linear part with small Lipschitz perturbation, then the covering relations must admit the weak Lyapunov condition and hence a subsystem has dynamics exactly as subshift dynamics. Moreover, we give the tolerance of Lipschitz perturbations controlled by the dichotomy tuple.

2. Statements of definitions and main results. First, we introduce some basic notations and definitions. For a positive integer K, let RK denote the space of all K-tuples of real numbers, |·| be the Euclidean norm on RK , and let k · k denote the operator-norm on the space of linear transformations on RK induced by |·|. For x ∈ RK and r > 0, we denote BK (x, r) = {z ∈ RK : |z − x| < r}; for the particular case when x = 0 and r = 1, we write BK = BK (0, 1), that is, the open unit ball in RK . Moreover, for a subset M of RK , let M, int(M) and ∂M denote the closure, interior and boundary of S, respectively. A function from a set into itself is also called a map on the set. Let N denote the set of all non-negative integers, while Z denotes the set of all integers. Let R+ denote the non-negative real numbers. We briefly recall some definitions from [25] concerning covering relations. Definition 2.1 (See [25, Definition 6]). An h-set in RK is a quadruple consisting of the following data: • a non-empty compact subset M of RK , • a pair of numbers u(M), s(M) ∈ {0, 1, ..., k} with u(M) + s(M) = K, and K K u(M) s(M) u(M) • a homeomorphism cM : R → R = R × R with cM (M) = B × Bs(M). For simplicity, we will denote such an h-set by M, and say that the h-set M has unstable (respectively, stable) direction, if u 6= 0 (respectively, s 6= 0); furthermore, we use the following notations: M˜ = Bu(M) × Bs(M), M˜ − = ∂Bu(M) × Bs(M), M˜ + = Bu(M) × ∂Bs(M), − −1 ˜ − + −1 ˜ + the unstable boundary M = cM (M ), and the stable boundary M = cM (M ); we also writev ˜ = cM (v) for v ∈ M. A covering relation between two h-sets is defined as follows. Definition 2.2 (See[25, Definition 7]). Let M,N be h-sets in RK with u(M) = u(N) = u and s(M) = s(N) = s, f : M → Ru × Rs be a continuous function, and ˜ −1 ˜ u s f := cN ◦ f ◦ cM : M → R × R . We say M f-covers N, denoted by f M =⇒ N, if the following conditions are satisfied: 1. there exists a homotopy h : [0, 1]×M˜ → Ru ×Rs such that h([0, 1], M˜ −)∩N˜ = ∅, h([0, 1], M˜ ) ∩ N˜ + = ∅, and h(0, x˜) = f˜(˜x) for allx ˜ ∈ M˜ ; 2. there exists a function ψ : Ru → Ru such that ψ(∂Bu) ⊂ Ru\Bu and h(1, p, q) = (ψ(p), 0) for all p ∈ Bu and q ∈ Bs; and 3. the local Brouwer degree of ψ at 0 in Bu is non-zero; refer to [25, Appendix] for its properties. Next, we introduce non-autonomous discrete dynamical systems as follows. Let K (fn)n∈Z be a two-sided sequence of continuous maps on R . For any i ∈ Z and j −j j −1 0 j ∈ N with j > 0, we set fi = fi+j−1 ◦ · · · ◦ fi+1 ◦ fi, fi = (fi ) , and fi = −1 id K , where the notation (g) will be applied to sets without assuming that g is R 5014 MING-CHIA LI AND MING-JIEA LYU

K invertible, and id K denotes the identity map on R . If fn = f for all n ∈ Z, then R j (fn)n∈Z = (f)n∈Z, the identical sequence. In general, fi is not the composition of fi with itself for j times, unless it is an identical sequence. A two-sided sequence of K compact subsets (Λn)n∈Z of a bounded set in R is said to be invariant for (fn)n∈Z if fn(Λn) = Λn+1 for all n ∈ Z. Similarly, replacing Z by N and two-sided sequence by one-sided sequence, one can consider one-sided non-autonomous discrete dynamical systems. From now on, the index, n ∈ N or n ∈ Z, of a sequence indicates the sequence is one-sided or two-sided, respectively. Notice that for invariance, we require the union of Λn’s to be bounded, the most interesting non-trivial dynamics. By a transition matrix, it means that a matrix satisﬁes (i) all entries are either zero or one, and (ii) all row sums and column sums are greater than or equal to one. + For a transition matrix A, let ΣA (respectively, ΣA) be the space of all two-sided (respectively, one-sided) allowable sequences for the matrix A with a usual metric, + + + and let σA :ΣA → ΣA (resp., σA :ΣA → ΣA) be the two-sided (respectively, one-sided) subshift of ﬁnite type induced by A; refer to [20] for more background.

For t ∈ ΣA, we write t = (ti)i∈Z. We deﬁne topological semi-conjugacy and topological conjugacy between non- autonomous systems and subshifts of ﬁnite type.

K Deﬁnition 2.3. Let (fn)n∈Z be a sequence of continuous maps on R , (Λn)n∈Z be a two-sided sequence of bounded subsets of which is invariant for (fn)n∈Z, and A be a transition matrix. We say that (fn)n∈Z restricted to (Λn)n∈Z is topologically semi-conjugate to the subshift of ﬁnite type σA, if there exists a sequence (ϕn)n∈Z of continuous functions such that for each n ∈ Z, ϕn maps Λn onto ΣA and ϕn+1 ◦ fn(x) = σA ◦ ϕn(x) for all x ∈ Λn, that is, for all n ∈ Z, the following diagrams commute:

fn−1 fn fn+1 fn+2 ··· −→ Λn −→ Λn+1 −→ Λn+2 −→ · · · · · · ↓ ϕn ↓ ϕn+1 ↓ ϕn+2 ··· · · · −→ ΣA −→ ΣA −→ ΣA −→ ··· σA σA σA σA

Moreover, we say that (fn)n∈Z restricted to (Λn)n∈Z is topologically conjugate to the subshift of ﬁnite type σA, if (fn)n∈Z is topologically semi-conjugate to σA, and for each n ∈ Z, ϕn is one-to-one on Λn and has continuous inverse. In the following, we deﬁne covering relations with a weak Lyapunov condition.

Deﬁnition 2.4. Let A = [aij]1≤i,j≤η be a transition matrix and (fn)n∈Z be a K two-sided sequence of continuous maps on R . We say that (fn)n∈Z has covering relations with a weak Lyapunov condition determined by A if there exist η h-sets η K {Mi}i=1 in R with pairwise disjoint interiors such that the following conditions are satisﬁed:

1. whenever aij = 1 and n ∈ Z,

fn Mi =⇒ Mj

and if a`j = ajγ = 1 and ` 6= i, then

fn(int(Mi)) ∩ fn(int(M`)) ∩ Mj = ∅, (2)

fn+1(fn(int(Mi) ∩ Mj)) ∩ fn+1(fn(int(M`)) ∩ Mj) ∩ Mγ = ∅; (3)

2. for each t ∈ ΣA, TOPOLOGICAL CONJUGACY FOR NON-AUTONOMOUS SYSTEMS 5015

(a) there exist two sequences (Pn[t])n∈Z and (Rn[t])n∈Z of real-valued func- u s tions on R and R respectively, such that Pn[t](x) > 0 and Rn[t](y) > 0 for all x 6= 0 and y 6= 0 and Pn[t](0) = Rn[t](0) = 0 for all n ∈ Z, where u denotes the dimension of the unstable direction of the covering relations and s = K − u;

(b) there exists a sequence (ξn[t])n∈Z of positive real numbers such that

Pn[t](x) ≤ ξn[t]|x| if n ≥ 0 and Rn[t](y) ≤ ξn[t]|y| if n < 0 (4) for all x ∈ Bu and y ∈ Bs;

(c) there exists a sequence (θn[t])n∈Z of non-negative real numbers such that for each n ∈ Z and for any two distinct points v, w in Mtn , if Pn[t](πu(˜v − w˜)) ≥ Rn[t](πs(˜v − w˜)) then ˜ ˜ ˜ ˜ Pn+1[t](πu(fn(˜v) − fn(w ˜))) ≥ Rn+1[t](πs(fn(˜v) − fn(w ˜))), (5) ˜ ˜ Pn+1[t](πu(fn(˜v) − fn(w ˜))) > (1 + θn[t])Pn[t](πu(˜v − w˜)) provided n ≥ 0, (6) ˜ ˜ Pn+1[t](πu(fn(˜v) − fn(w ˜))) > Pn[t](πu(˜v − w˜)) provided n < 0, (7)

and if Pn[t](πu(˜v − w˜)) ≤ Rn[t](πs(˜v − w˜)) then

Pn−1[t](πu(˜v−1 − w˜−1)) ≤ Rn−1[t](πs(˜v−1 − w˜−1)), (8)

Rn[t](πs(˜v − w˜)) < (1 − θn−1[t])Rn−1[t](πs(˜v−1 − w˜−1))

and θn−1 < 1 provided n ≤ 0, (9)

Rn[t](πs(˜v − w˜)) < Rn−1[t](πs(˜v−1 − w˜−1)) provided n > 0, (10) ˜ ˜ ˜ wherev ˜−1, w˜−1 are in Mtn−1 such that fn−1(˜v−1) =v ˜, fn−1(w ˜−1) =w ˜, u s u u s s and πu : R × R → R , πs : R × R → R are the projections defined u s by πu(x, y) = x, πs(x, y) = y for (x, y) ∈ R × R ; (d) whenever the h-sets have unstable direction, n−1 Y −1 lim ξn[t] (1 + θi[t]) = 0; (11) n→∞ i=0 and whenever the h-sets have stable direction, n Y lim ξn[t] (1 − θi[t]) = 0. (12) n→−∞ i=−1 Item 1 of the above definition indicates that the first two iterates of two points from interiors of distinct h-sets do not collapse at the same point in an h-set, which is in a sense of local injection. Item 2 demonstrates a weak Lyapunov condition, which is adapted from the cone condition for a covering relation given in [12, 24]; therein, Pn and Rn are required to be quadratic forms. Also Refer to Lyapunov function in [14] and Lyapunov sequences in [3]. Item 2 (d) is adapted from the good rate in our previous work [16, Definition 7]; therein, each ξn[t] is assumed to be one. The definition still make sense if there is no stable direction; in this case, one neglects the parts involving Rn’s and remains concerned (6), (7) and (11). It is similar for the the case when there is no unstable direction. Now, we present the first result that topological conjugacy is guaranteed by covering relation with a weak Lyapunov condition. K Theorem 2.5. Let (fn)n∈Z be a sequence of continuous maps on R which has covering relations with a weak Lyapunov condition determined by a transition matrix 5016 MING-CHIA LI AND MING-JIEA LYU

K A. Then there exists a sequence (Λn)n∈Z of compact subsets of R such that (Λn)n∈Z is invariant for (fn)n∈Z, and the restriction of (fn)n∈Z to (Λn)n∈Z is topologically conjugate to σA. In order to construct a weak Lyapunov condition for systems including nonhyperbolic ones, we recall the exponential dichotomy for a sequence of invertible matrices.

u Deﬁnition 2.6 (See[2]). Let (Dn)n∈Z be a sequence of invertible matrices on R × Rs of the following form Un 0 Dn = , 0 Sn u s where Un and Sn are matrices on R and R , respectively for each n ∈ Z. We say that (Dn)n∈Z admits an exponential dichotomy (more precisely, uniformly exponential dichotomy and non-uniformly exponential dichotomy, respectively) if there exist constants a < 0 < b, λ ≥ 1, and ε ≥ 0 (resp. ε = 0 and ε > 0) such that ε < min{|a|, b} and for every l, n ∈ Z with l ≤ n, kU(n, l)−1k ≤ λe−b(n−l)eε|n| and kS(n, l)k ≤ λea(n−l)eε|l|, (13) where   Wn−1 ··· Wm, n > m, W (n, m) = the identity matrix, n = m, for W = U or S. (14)  −1 −1 Wn ··· Wm−1, n < m, In this case, we call the tuple (a, b, λ, ε) the dichotomy tuple. In the following, we deﬁne the covering relations with exponential dichotomy determined by a transition matrix.

Definition 2.7. Let A = [aij]1≤i,j≤η be a transition matrix and (fn)n∈Z be a K sequence of continuous maps on R . We say that (fn)n∈Z has covering relations η K with exponential dichotomy determined by A if there exist η h-sets {Mi}i=1 in R with pairwise disjoint interiors such that the following conditions are satisfied: 1. item 1 of Definition 2.4 is satisfied; 2. for any allowable sequence t in Σ , the local map f˜ [t t ] := c ◦ f ◦ A n n n+1 Mtn+1 n c−1 is of the form Mtn ˜ ˜ fn[tntn+1] = Dn[tntn+1] + gn[tntn+1] on Mtn , (15) K where Dn[tntn+1] is a linear map and gn[tntn+1] is a continuous map on R such that (a) Dn[tntn+1] is decoupled as follows Un[tntn+1] 0 Dn[tntn+1] = (16) 0 Sn[tntn+1] u s for some matrices Un[tntn+1], Sn[tntn+1] on R and R , respectively, where u denotes the dimension of the unstable direction of the covering relations and s = K − u; (b) gn[tntn+1] is Lipschitz with some Lipschitz constant, namely Ln[tntn+1], as follows ˜ |gn[tntn+1](˜v) − gn[tntn+1](w ˜)| ≤ Ln[tntn+1]|v˜ − w˜| for allv ˜,w ˜ in Mtn ; (17) TOPOLOGICAL CONJUGACY FOR NON-AUTONOMOUS SYSTEMS 5017

3. there exists a tuple (a, b, ε, λ) such that for each allowable sequence t in ΣA,

the sequence (Dn[tntn+1])n∈Z of linear maps admits exponential dichotomy with the dichotomy tuple (a, b, λ, ε). The dichotomy tuple of the linear part allows us to determine a tolerance for stability. Let ρ and ρ0 are two real numbers such that 0 < ρ < ρ0 < min{|a|, b} − ε. (18) We set the tolerance as follows: for each n ∈ Z,

 0 0  1 − e−ρ  eb−ρ − eε e−ε − ea+ρ eb−ρ − 1 1 − ea+ρ  δ = min , , , . (19) n ε|n+1| q q λe  λeε|n| 2 λeε|n| 2 2 2  1 + ( 1−e−ρ ) 1 + ( 1−e−ρ ) Now, we present the second result that if the Lipschitz constant of the non- linear part is less than tolerance, then there is way to construct a weak Lyapunov condition, and hence, as an application of Theorem 2.5, there exists a subsystem which is topologically conjugate to a subshift of ﬁnite type.

Theorem 2.8. Let (fn)n∈Z be a sequence of continuous maps which has covering relations with exponential dichotomy determined by a transition matrix A. If for any integer n and any allowable sequence t in ΣA the Lipschitz constant Ln[tntn+1] in (17) is less than the tolerance δn in (19), then (fn)n∈Z has covering relations with a weak Lyapunov condition determined by a transition matrix A. Notice that since the dimension of the transition matrix A is ﬁnite, namely η, 2 for each n ∈ Z, there are at most η Lipschitz constants Ln[tntn+1] in (17) for all allowable sequences t in ΣA, and hence there exists the maximum; which is independent of n if the systems involves only ﬁnitely many fn’s, e.g. the identity sequence (f)n∈Z. Moreover, in the case of uniformly exponential dichotomy, the tolerance in (19) becomes independent of n as follows:

 0 0  1 − e−ρ  eb−ρ − 1 1 − ea+ρ eb−ρ − 1 1 − ea+ρ  δn = min q , q , , . λ λ 2 λ 2 2 2  1 + ( 1−e−ρ ) 1 + ( 1−e−ρ )  At the end, we state a result which extends [16, Theorem 2] by removing con- strains on quadratic form, continuity, and ﬁniteness of real-valued functions con- trolling the Lyapunov condition. Theorem 2.9. The statement of Theorem 2.5 remains valid if item 2 (a)-(c) of Deﬁnition 2.4 is replaced as follows: K (a’) there exists a sequence {Qn[t]}n∈Z of real-valued functions on R such that for any n ∈ Z, u s Qn[t](x, y) = Pn[t](x) − Rn[t](y) for all (x, y) ∈ R × R , (20) u s where Pn[t] and Rn[t] are real-valued functions on R and R respectively such that Pn[t](x) > 0 and Rn[t](y) > 0 for all x 6= 0 and y 6= 0 and Pn[t](0) = Rn[t](0) = 0, where u denotes the dimension of the unstable direction of the covering relations and s = K − u;

(b’) there exists a sequence (ξn[t])n∈Z of positive real numbers such that

|Qn[t](z)| ≤ ξn[t]|z| (21) ˜ for all n ∈ Z and z ∈ Mtn ; and 5018 MING-CHIA LI AND MING-JIEA LYU

(c’) there exist a sequence (θn[t])n∈Z of non-negative real numbers such that for any n ∈ Z and any two distinct points v, w in Mtn , if Qn[t](˜v − w˜) ≥ 0 then ˜ ˜ Qn+1[t](fn(˜v) − fn(w ˜)) > (1 + θn[t])Qn[t](˜v − w˜) provided n ≥ 0, (22) ˜ ˜ Qn+1[t](fn(˜v) − fn(w ˜)) > (Qn[t](˜v − w˜) provided n < 0,

and if Qn[t](˜v − w˜) < 0 then

Qn[t](˜v − w˜) > (1 − θn−1[t])Qn−1[t](˜v−1 − w˜−1) and θn−1 < 1 provided n ≤ 0, (23)

Qn[t](˜v − w˜) > Qn−1[t](˜v−1 − w˜−1) provided n > 0, ˜ ˜ ˜ where v˜−1, w˜−1 are in Mtn−1 such that fn−1(˜v−1) =v ˜ and fn−1(w ˜−1) =w ˜. For one-sided systems, all above definitions and results still valid under slight modifications as follows: + + • in all definitions and theorems, replace Z by N,ΣA by ΣA, and σA by σA . • in Definition 2.4 and 2.7, add the h-sets are pairwise disjoint and neglect (2) and (3). • in Theorems 2.5, 2.8 and 2.9, add the assumption that the h-sets has no stable direction.

3. Proof of results. Through out this section, we let (fn)n∈Z be a non-autonomous system having covering relations determined by the transition matrix A = η [aij]1≤i,j≤η on the pairwise disjoint h-sets {Mi}i=1. For t ∈ ΣA, we write t = (ti)i∈Z. K First, we construct a sequence (Λn)n∈Z of compact subsets of R such that (Λn)n∈Z is invariant for (fn)n∈Z, and the restriction of (fn)n∈Z to (Λn)n∈Z is topologically semi-conjugate to σA; here we give a brief outline of the proof and refer the readers to [16, Theorem 1] for more details. For n ∈ Z, define k \ i −i Λn,k,t = fn−i(Mt−i ) ∩ fn (Mti ) for k ∈ N and t ∈ ΣA, i=0 [ \ Λn,k = Λn,k,t for k ∈ N, and Λn = Λn,k. t∈ΣA k∈N Then, Λn is the set of all points whose orbits acted by fn at the beginning, Sη following allowable sequences in ΣA, and stay in i=1 Mi. Furthermore, Λn = S T η Λn,k,t . Thus, (Λn)n∈ is invariant for (fn)n∈ in ∪ Mi with re- t∈ΣA k∈N Z Z i=1 spect to A and each Λn is compact. For each n ∈ Z, we can define n ϕn :Λn → ΣA by ϕn(z) = σA(t), i where fn (z) ∈ Mtn+i for all i ∈ Z. It is not difficult to show that ϕn+1 ◦ fn(z) = σA ◦ ϕn(z) for all z ∈ Λn, and ϕn is continuous on Λn for all integer n. With a help T of [25, Theorem 9], one can claim that for any t ∈ ΣA, the intersection Λn,k,t k∈N is non-empty, which lead a conclusion that each ϕn is surjective on Λn.

Proof of Theorem 2.5. By the above, it suﬃces to show that ϕn is injective on Λn for all n ∈ Z. Let n ∈ Z and t ∈ ΣA be arbitrary. Suppose on contrary that T vn, wn ∈ Λn,i,t with vn 6= wn. For convenience, we denote by {vi}i∈ a (not i∈N Z necessarily unique) orbit of vn under (fi)i∈Z, that is, fi(vi) = vi+1 for all i ∈ Z; similarly, for wi. First, we consider the case when

Pn[t](πu(˜vn − w˜n)) ≥ Rn[t](πs(˜vn − w˜n)). TOPOLOGICAL CONJUGACY FOR NON-AUTONOMOUS SYSTEMS 5019

Sincev ñ 6=w ñ, the positivity of Pn[t] and Rn[t] implies Pn[t](πu(˜vn − wñ)) > 0. By using (6) and (7), we get Pn+1[t](πu(˜vn+1 −wñ+1)) > 0 and hence,v ñ+1 6=w ñ+1. In- ductively, we obtain that Pn+i[t](πu(˜vn+i − wñ+i)) > 0 for all non-negative integers Sη ∞ i. The compactness of the union i=1 Mi implies that both sequences {vn+i}i=0 ∞ ∞ ∞ and {wn+i}i=0 have convergent subsequences, say {vn+i(j)}j=0 and {wn+i(j)}j=0, η with the limits, say v∞ and w∞ in ∪i=1Mi, respectively. Let β be a non-negative integer such that n + i(β) ≥ 0. By using (4) and (6), we get that for all j ≥ β, −1 |πu(˜vn+i(j) − wñ+i(j))| ≥ (ξn+i(j)[t]) Pn+i(j)[t](πu(˜vn+i(j) − wñ+i(j))) n+i(j)−1  −1 Y > (ξn+i(j)[t])  (1 + θl[t]) Pn[t](πu(˜vn − wñ)). l=n+i(β)

Taking j → ∞, because of (11), we obtain |πu(˜v∞ − w˜∞)| ≥ ∞. This leads to a contradiction. Similarly, for the case when

Pn(πu(˜vn − wñ)) < Rn(πs(˜vn − wñ)), we have that Rn−i[t](πs(˜vn−i −wñ−i)) > 0 for all non-negative integers i. Moreover, ∞ ∞ both the sequences {vn−i}i=0 and {wn−i}i=0 have convergent subsequences, say ∞ ∞ Sη {vn−i(j)}j=0 and {wn−i(j)}j=0, with the limits, namely v−∞ and w−∞ in i=1 Mi, respectively. Let β be a non-negative integer such that n − i(β) < 0. By using (9) and (4), we get that for j ≥ β,

 n−i(j)  Y Rn[t](πs(˜vn − w˜n)) <  (1 − θl[t]) Rn−i(j)[t](πs(˜vn−i(j) − w˜n−i(j))) l=n−i(β)

 n−i(j)  Y ≤  (1 − θl[t]) ξn−i(j)[t]|πs(˜vn−i(j) − w˜n−i(j))|. l=n−i(β)

Taking j → ∞, because of (12), we obtain 0 < Rn[t](πs(˜vn − wñ)) ≤ 0; a contradiction. Since n ∈ Z is arbitrarily chosen, we have finished the proof of the theorem. Proof of Theorem 2.8. It is sufficient to check all conditions in Definition 2.4 are satisfied. It is clear that item (1) of Definition 2.4 is satisfied. For item (2), given an t ∈ ΣA, we first construct sequences of real-valued continuous functions (Pn[t])n∈Z u s u s and (Rn[t])n∈Z on R and R , respectively. For n ∈ Z and (x, y) ∈ R × R , we define X (b−ρ)(n−l) X −(a+ρ)(l−n) Pn[t](x) = |U(l, n)x|e and Rn[t](y) = |S(l, n)y|e . (24) l≤n l≥n From the exponential dichotomy in (13) and (14), we get |U(l, n)x|e(b−ρ)(n−l) ≤ λe−b(n−l)+ε|n|e(b−ρ)(n−l)|x| = λe−ρ(n−l)eε|n||x| for l ≤ n and |S(l, n)y|e−(a+ρ)(l−n) ≤ λea(l−n)+ε|n|e−(a+ρ)(l−n)|y| = λe−ρ(l−n)eε|n||y| for l ≥ n. By (18), we obtain X λeε|n| X λeε|n| |U(l, n)x|e(b−ρ)(n−l) ≤ |x| and |S(l, n)y|e−(a+ρ)(l−n) ≤ |y|. 1 − e−ρ 1 − e−ρ l≤n l≥n 5020 MING-CHIA LI AND MING-JIEA LYU

It follows that in (24), both real-valued functions Pn[t] and Rn[t] are well defined and continuous, and satisfy the conditions in item (2a) of Definition 2.4. Moreover, we have λeε|n| λeε|n| P [t](x) ≤ |x| and R [t](y) ≤ |y|. (25) n 1 − e−ρ n 1 − e−ρ For each n ∈ Z, set ε|n| −ρ −1 ξn[t] = λe (1 − e ) . (26) u s Then for (x, y) ∈ R × R , Pn[t](x) ≤ ξn[t]|x| for all n ≥ 0 and Rn[t](y) ≤ ξn[t]|y| for all n < 0. This verifies the conditions in item (2b) of Definition 2.4. For item (2c) of Definition 2.4, we split the proof into four steps. Step 1, we set the following notations: η = min{eb−ρ − 1, 1 − ea+ρ} (27) and for n ∈ Z, s λL [t t ]eε|n+1| λeε|n| 2 ϑ [t] = eb−ρ − 1 − n n n+1 1 + , (28) n 1 − e−ρ 1 − e−ρ s λL [t t ]eε|n+1| λeε|n| 2 ϑ0 [t] = 1 − ea+ρ − n n n+1 1 + , (29) n 1 − e−ρ 1 − e−ρ 0 θn[t] = ϑn[t] provided n ≥ 0, and θn[t] = ϑn[t] provided n < 0. (30)

By (18), (19), (27), and the hypothesis Ln[tntn+1] < δn, we get that for each n ∈ Z, 2λL [t t ]eε|n+1| η − n n n+1 > 0, (31) 1 − e−ρ −ε b−ρ b−ρ0 ε −ε b−ε −ρ −ρ0 (1 + ϑn[t])e > e − (e − e ) e = 1 + e (e − e ) > 1, (32)

0 ε a+ρ −ε a+ρ0 ε a+ε ρ0 ρ (1 − ϑn[t])e < e + (e − e ) e = 1 − e (e − e ) < 1. (33) Hence, ε 0 −ε ϑn[t] > e − 1 ≥ 0 and 1 > ϑn[t] > 1 − e ≥ 0. (34)

Step 2, we estimate the values of Pn+1[t] and Rn+1[t] for the iterate of the linear part Dn[tntn+1]. To simplify notation, we write Dn = Dn[tntn+1], Un = Un[tntn+1] and Sn = Sn[tntn+1]. Let v, w ∈ Mtn be arbitrary. By the linearity in (16), we have πu(Dnv˜ − Dnw˜) = Unπu(˜v − w˜) and πs(Dnv˜ − Dnw˜) = Snπs(˜v − w˜). By (24), (14), and (27), we get that

Pn[t](πu(˜v − w˜)) ≥ |πu(˜v − w˜)| and Rn[t](πs(˜v − w˜)) ≥ |πs(˜v − w˜)|, (35)

≥ (1 + η)Pn[t](πu(˜v − w˜)), (37) TOPOLOGICAL CONJUGACY FOR NON-AUTONOMOUS SYSTEMS 5021 and

≤ (1 − η)Rn[t](πs(˜v − w˜)). (39) Together with (37) and (39), it follows that

Pn+1[t](πu(Dn(˜v) − Dn(w ˜))) − Rn+1[t](πs(Dn(˜v) − Dn(w ˜)))

≥ Pn[t](πu(˜v − w˜)) − Rn[t](πs(˜v − w˜)) + η (Pn[t](πu(˜v − w˜)) + Rn[t](πs(˜v − w˜)))

≥ Pn[t](πu(˜v − w˜)) − Rn[t](πs(˜v − w˜)) + η (|πu(˜v − w˜)| + |πs(˜v − w˜)|)

≥ Pn[t](πu(˜v − w˜)) − Rn[t](πs(˜v − w˜)) + η|v˜ − w˜|. (40)

Step 3, we estimate the values of Pn+1[t] and Rn+1[t] for the non-linear maps ˜ ˜ ˜ fn[tntn+1]. Again, to simplify notation, we write the non-linear map fn = fn[tntn+1], the non-linear part gn = gn[tntn+1], and the components gn(z) = (gn,1(z), gn,2(z)) ∈ Ru × Rs for z ∈ Ru × Rs. By (15) and (16), ˜ ˜ fn(˜v) − fn(w ˜) = (Unπu(˜v − w˜) + gn,1(˜v) − gn,1(w ˜),Snπs(˜v − w˜) + gn,2(˜v) − gn,2(w ˜)). By the triangle inequality, (13) and (17), we have ˜ ˜ Pn+1[t](πu(fn(˜v) − fn(w ˜)))

= Pn+1[t](Unπu(˜v − w˜) + gn,1(˜v) − gn,1(w ˜)) X (b−ρ)(n+1−l) = |U(l, n + 1)(Unπu(˜v − w˜) + gn,1(˜v) − gn,1(w ˜))|e l≤n+1 X (b−ρ)(n+1−l) ≥ Pn+1[t](πu(Dnv˜ − Dnw˜)) − kU(l, n + 1)kLn[tntn+1]e |v˜ − w˜| l≤n+1 ε|n+1| −ρ(n+1) X ρl ≥ Pn+1[t](πu(Dnv˜ − Dnw˜)) − λLn[tntn+1]e e e |v˜ − w˜| l≤n+1 λL [t t ]eε|n+1| = P [t](π (D v˜ − D w˜)) − n n n+1 |v˜ − w˜|. (41) n+1 u n n 1 − e−ρ Similarly, we get ˜ ˜ Rn+1[t](πs(fn(˜v) − fn(w ˜)))

= Rn+1[t](Snπs(˜v − w˜) + gn,2(˜v) − gn,2(w ˜)) X −(a+ρ)(l−n−1) = |S(l, n + 1)(Snπs(˜v − w˜) + gn,2(˜v) − gn,2(w ˜))|e l≥n+1 X −(a+ρ)(l−n−1) ≤ Rn+1[t](πs(Dnv˜ − Dnw˜)) + ||S(l, n + 1)||Ln[tntn+1]e |v˜ − w˜| l≥n+1 ε|n+1| −ρ(n+1) X ρl ≤ Rn+1[t](πs(Dnv˜ − Dnw˜)) + λLn[tntn+1]e e e |v˜ − w˜| l≥n+1 5022 MING-CHIA LI AND MING-JIEA LYU

λL [t t ]eε|n+1| = R [t](π (D v˜ − D w˜)) + n n n+1 |v˜ − w˜|. (42) n+1 s n n 1 − e−ρ

Combining (40)-(42), and using (31), we obtain

˜ ˜ ˜ ˜ Pn+1[t](πu(fn(˜v) − fn(w ˜))) − Rn+1[t](πs(fn(˜v) − fn(w ˜)))

≥ Pn+1[t](πu(Dnv˜ − Dnw˜)) − Rn+1[t](πs(Dnv˜ − Dnw˜)) 2λL [t t ]eε|n+1| − n n n+1 |v˜ − w˜| 1 − e−ρ 2λL [t t ]eε|n+1| ≥ P [t](π (˜v − w˜)) − R [t](π (˜v − w˜)) + η − n n n+1 |v˜ − w˜| n u n s 1 − e−ρ (43)

≥ Pn[t](πu(˜v − w˜)) − Rn[t](πs(˜v − w˜)). (44)

Step 4, we are in position to verify item (2c) of Deﬁnition 2.4. Let n ∈ Z and let v, w be any two distinct points in Mtn . First, we consider the case when Pn[t](πu(˜v − w˜)) ≥ Rn[t](πs(˜v − w˜)). Since v 6= w, the non-negativity of Pn[t] and Rn[t] implies Pn[t](πu(˜v − w˜)) > 0 and hence, πu(˜v − w˜) 6= 0. By (36),

b−ρ Pn+1[t](πu(Dnv˜ − Dnw˜)) > e Pn[t](πu(˜v − w˜)). (45)

On the other hand, by (25) and (35), we get

p 2 2 p 2 2 |v˜ − w˜| = |πu(˜v − w˜)| + |πs(˜v − w˜)| ≤ |πu(˜v − w˜)| + (Rn[t](πs(˜v − w˜))) s λeε|n| 2 ≤ p|π (˜v − w˜)|2 + (P [t](π (˜v − w˜)))2 ≤ 1 + |π (˜v − w˜)| u n u 1 − e−ρ u s λeε|n| 2 ≤ 1 + P [t](π (˜v − w˜)). (46) 1 − e−ρ n u

Using (41), (46), (45), and (28), we obtain

˜ ˜ Pn+1[t](πu(fn(˜v) − fn(w ˜))) λL [t t ]eε|n+1| ≥ P [t](π (D v˜ − D w˜)) − n n n+1 |v˜ − w˜| n+1 u n n 1 − e−ρ s λL [t t ]eε|n+1| λeε|n| 2 > eb−ρP [t](π (˜v − w˜)) − n n n+1 1 + P [t](π (˜v − w˜)) n u 1 − e−ρ 1 − e−ρ n u

= (1 + ϑn[t])Pn[t](πu(˜v − w˜)).

Therefore, (6) follows (34) and hence, (7) follows (34). Apparently, (5) follows (44). Next, we consider the case when Pn[t](πu(˜v − w˜)) ≤ Rn[t](πs(˜v − w˜)). Since ˜ ˜ v˜ 6=w ˜, we havev ˜−1 6=w ˜−1 where fn−1(˜v−1) =v ˜, fn−1(w ˜−1) =w ˜. Using (31) and (43), we get Pn−1[t](πu(˜v−1 − w˜−1)) < Rn−1[t](πs(˜v−1 − w˜−1)) and hence, πs(˜v−1 − w˜−1) 6= 0. By (38),

a+ρ Rn[t](πs(Dn−1v˜−1 − Dn−1w˜−1)) < e Rn−1[t](πs(˜v−1 − w˜−1)). (47) TOPOLOGICAL CONJUGACY FOR NON-AUTONOMOUS SYSTEMS 5023

By (25) and (35), we get

|v˜−1 − w˜−1| p 2 2 = |πu(˜v−1 − w˜−1)| + |πs(˜v−1 − w˜−1)| p 2 2 ≤ (Pn−1[t](πu(˜v−1 − w˜−1))) + |πs(˜v−1 − w˜−1)| p 2 2 ≤ (Rn−1[t](πs(˜v−1 − w˜−1))) + |πs(˜v−1 − w˜−1)| s λeε|n−1| 2 ≤ 1 + |π (˜v − w˜ )| 1 − e−ρ s −1 −1 s λeε|n−1| 2 ≤ 1 + R [t](π (˜v − w˜ )). (48) 1 − e−ρ n−1 s −1 −1

Using (42), (48), (47), and (29), we obtain

Rn[t](πs(˜v − w˜)) λL [t t ]eε|n| ≤ R [t](π (D v˜ − D w˜ )) + n−1 n−1 n |v˜ − w˜ | n s n−1 −1 n−1 −1 1 − e−ρ −1 −1 a+ρ < e Rn−1[t](πs(˜v−1 − w˜−1)) s λL [t t ]eε|n| λeε|n−1| 2 + n−1 n−1 n 1 + R [t](π (˜v − w˜ )) 1 − e−ρ 1 − e−ρ n−1 s −1 −1 0 = (1 − ϑn−1[t])Rn−1[t](πs(˜v−1 − w˜−1)). Therefore, (9) follows (34) and hence, (10) follows (34). Apparently, (8) follows (44). We have shown that all the conditions in item (2c) of Theorem 2.5 are satisﬁed. It remains to verify in item (2d) of Theorem 2.5. Combining (26), (30), (32), and (33), we get that for each m > 0,

m−1 m−1 Y λ Y 1 0 < ξ [t] (1 + θ [t])−1 < m l 1 − e−ρ 1 + eb−ε(e−ρ − e−ρ0 ) l=0 l=0 and −m −m Y λ Y 0 0 < ξ [t] (1 − θ [t]) < 1 − ea+ε(eρ − eρ) ; −m l 1 − e−ρ l=−1 l=−1 moreover, both terms on the right-hand sides tend to zero as m goes to the inﬁnity, and hence,

m−1 −m Y −1 Y lim ξm[t] (1 + θl[t]) = lim ξ−m[t] (1 − θl) = 0. m→∞ m→∞ l=0 l=−1 We have ﬁnished the proof of the theorem.

The proof of Theorem 2.9 is very similar to the proof of Theorem 2.5 by splitting into two cases: Qn[t] is non-negative and is negative; refer to (20). We omit it here. 5024 MING-CHIA LI AND MING-JIEA LYU

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