Covering relations determined by a transition matrix . 43

Here, we will state the definition of the covering relations determined by a transition matrix and list the related main results of the topological dynamics for multidimensional perturbations of high-dimensional systems.

First, we introduce the transition matrix. By a transition matrix, it means that a square matrix satisfies (i) all entries are either zero or one, and (ii) all row sums and column sums are are greater than or equal to one. For a transition matrix A, let ρ(A) denote the spectral radius of A. Then ρ(A) > 1 and moreover, if A is irreducible and not a permutation, then ρ(A) > 1. Let

Σ⁺_A (resp. Σ_A) be the space of all allowable one-sided (resp. two sided) sequences for the matrix A with a usual metric, and let σ⁺_A : Σ⁺_A → Σ⁺_A (resp. σ_A : Σ_A → Σ_A) be the one-sided (resp. two sided) subshift of finite type for A. Then h_top(σ_A⁺) = h_top(σ_A) = log(ρ(A)). Refer to [29] for more background.

Next, we define covering relation determined by a transition matrix.

Definition 4.21. Let A = [aij]_16i,j6γ be a transition matrix and f be a continuous map on R^m. We say that f has covering relations determined by A if the following conditions are satisfied:

1. there are γ pairwise disjoint h-sets {M_i}^γ_i=1 in R^m; 2. if a_ij = 1 then the covering relation M_i =⇒ M^f _j holds.

It is easy to see that the logistic maps f (x) = µx(1 − x) with µ > 4 has covering relations determined by the 2 × 2 matrix with all entries one on intervals [−, 1/2 − δ] and [1/2 + δ, 1 + ] as h-sets, where 0 <  < µ/4 − 1 and 0 < δ < [(µ/4 − 1 − )µ⁻¹]^1/2.

Now, we begin to state the main theorems of covering relation determined by a transition matrix.

Theorem 4.22. Let f be a continuous map on R^m having covering relations determined by a transition matrix A. If g is a continuous map on R^m with

|g − f | small enough, then h_top(g) > log(ρ(A)).

If the singular map F depends only on the phase variable of f , we have the following result about multidimensional perturbations.

Theorem 4.23. Let F (x, y) = (f (x), g(x)) ∈ R^m× R^k for all x ∈ R^m and y ∈ R^k, where f : R^m → R^m is a continuous map having covering relations determined by a transition matrix A, and g : R^m → R^k is a continuous function. If G is a continuous map on R^m× R^k with |G − F | small enough, then h_top(G) > log(ρ(A)).

For the case when the singular map is a skew product locally trapping along the second variable, we have the following.

Theorem 4.24. Let F (x, y) = (f (x), g(x, y)) ∈ R^m × R^k for all x ∈ R^m and y ∈ R^k, where f : R^m → R^m is a continuous map having covering relations determined by a transition matrix A, and g : R^m × R^k → R^k is a continuous function such that g(R^m × S) ⊂ int(S) for some compact set S ⊂ R^k homeomorphic to the closed unit ball in R^k. If G is a continuous map on R^m× R^k with |G − F | small enough, then h_top(G) > log(ρ(A)).

In order to prove the above results, we need a proposition which is de-scribed that a continuous map having covering relations determined by a transition matrix is topologically semi-conjugate to a one-sided subshift of finite type. A variant version of the this result was stated without proof in [7, Corollary 5.9].

Proposition 4.25. Let f : R^m → R^m be a continuous map which has cover-ing relations determined by a transition matrix A. Then there exists a com-pact subset Λ of R^m such that Λ is maximal positive invariant for f in the union of the h-sets (with respect to A) and f |Λ is topologically semi-conjugate to σ_A⁺.

Proof. For convenience, we denote by {M_i}^η_i=1 the h-sets with covering re-lations for f determined by A as in Definition 4.21, and write we write s = (s₀, s₁, . . .) for s ∈ Σ⁺_A. Define

Then Λ is the set of all points whose forward orbits, following allowable sequences in Σ⁺_A, stay in ∪^η_i=1M_i. Thus Λ is maximal positive invariant set for f in ∪^η_i=1M_i with respect to A. Since each M_i is compact and f is continuous, the set ∩ⁿ_i=0f⁻ⁱ(M_si) is compact for all n > 0 and s ∈ Σ⁺A. Since

the number of sets M_i’s is η and the intersection ∩ⁿ_i=0f⁻ⁱ(M_si) involves only the first n + 1 digits of s ∈ Σ⁺_A, that is, (s₀, s₁, . . . , s_n), there are at most ηⁿ⁺¹ sets ∩ⁿ_i=0f⁻ⁱ(M_si) for all s ∈ Σ⁺_A, although the set Σ⁺_A itself might be uncountable. Thus the set Λ_n is a union of finitely many compact sets and hence is compact for all n > 0. Therefore, Λ is compact.

For semi-conjugacy, we define h : Λ → Σ⁺_A by h(z) = s for z ∈ Λ, where fⁿ(z) ∈ M_sn for all n > 0. By the pairwise disjointness of Mi’s and the definition of Λ, the map h is well defined. It is easy to show that σ_A◦ h = h ◦ f . Next, we show that h is continuous on Λ. Let z ∈ Λ, h(z) = s and {z_n}^∞_n=1 be a sequence in Λ such that z_n → z as n → ∞. Since M_i’s are pairwise disjoint and compact, there exists n₀ ∈ N such that zn∈ M_s0 for all n > n0. By the continuity of f , there exists n₁ ∈ N such that n1 > n0 and f (z_n) ∈ M_s1 for all n > n1. By using the same process inductively, we get that for each i > 0, there exist there exists ni ∈ N such that f^j(z_n) ∈ M_sj for all n > ni and 0 6 j 6 i. This proves that h(zn) → s as n → ∞. Therefore, h is continuous on Λ.

To prove that h is onto, we need the following lemma.

Lemma 4.26. For any s ∈ Σ_A, the intersection ∩_n>0f⁻ⁿ(M_sn) is nonempty.

Proof. Let s ∈ Σ_A. First, we prove that the intersection ∩ⁿ_i=0f⁻ⁱ(M_si) is nonempty for all n > 0 by applying Theorem 3.3 to a closed loop of covering relations. Let n > 0. Then we have the loop of covering relations Ms0

=⇒f

M_s1 =⇒ · · ·^f =⇒ M^f _sn. The loop becomes closed by adding a covering relation M_sn =⇒ M^g _s0 with a homotopy h : [0, 1]×M_sn,c → R^u×R^s, where u = u(M_sn), s = s(M_sn), g_c : R^u × R^s → R^u × R^s is defined by g_c(p, q) = (2p, 0) for all (p, q) ∈ R^u × R^s, g = c⁻¹_M

s0 ◦ g_c ◦ c_Msn, and h(t, p, q) = g_c(p, q) for t ∈ [0, 1] and (p, q) ∈ M_sn,c. It follows from Theorem 3.3 that there exists z ∈ int(M_s0) such that fⁱ(z) ∈ int(M_si) for 0 6 i 6 n. Thus z ∈ ∩ⁿi=0f⁻ⁱ(M_si).

Therefore, the intersection ∩ⁿ_i=0f⁻ⁱ(M_si) is nonempty for all n > 0. Since

{∩ⁿ_i=0f⁻ⁱ(M_si)}^∞_n=0 is a nested sequence of nonempty compact subsets of R^m,

Now, we begin to prove the results for covering relation determined by a transition matrix. In the following, we write A = [a_ij]_16i,j6η and denote by {M_i}^η_i=1 the pairwise disjoint h-sets with covering relations for f determined by A.

Proof of Theorem 4.22. Since the dimension of A is η, there are at most η² choices of the covering relations M_i =⇒ M^f _j. By Proposition 3.5, if g : R^m → R^m is a continuous map with |g − f | small enough, then g has covering rela-tions on h-sets {M_i}^η_i=1determined by A. By applying Proposition 4.25 to the map g, there exists a compact subset Λ_g of R^m such that Λ_g is positive invari-ant for g and g|Λ_g is topologically semi-conjugate to the one-sided subshift of finite type σ⁺_A. Therefore, h_top(g) > htop(g|Λ_g) > htop(σ_A⁺) = log(ρ(A)).

where h is the homotopy for the covering relation M_i =⇒ M^f _j. This shows that F has covering relations on {M_i⁰}^η_i=1 determined by A. By applying

Theorem 4.22 to the map F on R^m× R^k, we get that if G is a continuous map on R^m × R^k with |G − F | small enough, then there exists a compact subset Λ_G of R^m+k such that Λ_G is positively invariant for g and g|Λg is topologically semi-conjugate to the one-sided subshift of finite type σ⁺_A. Proof of Theorem 4.24. Define ˆF = (id, c) ◦ F ◦ (id, c)⁻¹, where id denotes the identity map on R^m and c is a homeomorphism form S to B_k. Then the conclusion follows from the above argument.

4.3.2 Liapunov and strong Liapunov condition

In this subsection, we will introduce covering relations with the the Liapunov and strong Liapunov conditions determined by a transition matrix and list the related main results of the topological dynamics for multidimensional perturbations of high-dimensional systems. Here, we will let | · | denote the Euclidean norm and || · || denote the operator norm on the space of linear maps induced by | · |.

In the following, we slightly modify the cone condition for a covering relation given by Zgliczy´nski in [36, Definition 11] and furthermore, we define the strong Liapunov condition. First, We define a quadratic form on a h-set K in R^m to be of the form

Q_K(x, y) = P_K(x) − Q_K(y) for all (x, y) ∈ R^u(K)× R^s(K), (4.20) where P_K : R^u(K) → R and QK : R^s(K) → R are positive definite quadratic forms. Note that a quadratic form on Rⁿ is a function Q defined on Rⁿ whose value at a vector z in Rⁿ can be computed by an expression of the form Q(z) = z^TSz, where S is an n × n symmetric matrix and z^T denotes the transpose of z; refer to [27].

Definition 4.27. Let Q_M and Q_N be quadratic forms on h-sets M and N , respectively, as in Equation (4.20). We say that a covering relation M =⇒ N^f

satisfies the Liapunov condition (resp. the strong Liapunov condition) with respect to the pair (Q_M, Q_N) if there exists θ > 0 (resp. θ > 0) such that for any u, v ∈ M_c with u 6= v,

QN(fc(u) − fc(v)) − QM(u − v) > θ |u − v|². (4.21) As a Liapunov function, a sequence of quadratic forms has scalar values strictly monotone along the difference of two orbits. More precisely, consider covering relations Mi

=⇒ Mf i+1 satisfying the Liapunov condition with re-spect to the pair (Q_Mi, Q_Mi+1) of quadratic forms for all i > 0. If u, v are two points such that fⁱ(u), fⁱ(v) ∈ M_i,c and fⁱ(u) 6= fⁱ(v) for all i > 0, then the sequence {QMi(fⁱ(u) − fⁱ(v)}^∞_i=0is strictly increasing. This property will play an import role while we prove conjugacy results.

Definition 4.28. Let A = [a_ij]_16i,j6η be a transition matrix and f be a continuous map on R^m. We say that f has covering relations with the Lia-punov conditions (resp. the strong LiaLia-punov condition) determined by A if the following conditions are satisfied;

1. there are η pairwise disjoint h-sets {M_i}^η_i=1 in R^m; on each M_i there exists a quadratic form Q_Mi as in Equation (4.20).

2. if aij = 1 then the covering relation Mi

=⇒ Mf j holds and satisfies the Liapunov condition (resp. the strong Liapunov condition) with respect to the pair (Q_Mi, Q_Mj); and

3. if aij = 1 then the coordinate chart cMi and cMj is a C¹ diffeomor-phisms.

The Liapunov condition is for detection of chaos (see Proposition 4.33 below), while the strong Liapunov condition is for stability of chaos under small C¹ perturbations as follows.

Next, we use the logistic map once again as an example of map has covering relations with the Liapunov conditions (resp. the strong Liapunov condition) determined by a transition matrix.

Example 4.29. Let us show that the logistic map f (x) = µx(1 − x) with µ > 4 has covering relations with the strong Liapunov condition determined by the 2 × 2 matrix with all entries one. Set (i) h-sets M₁ = [−, 1/2 − δ] and lemma and the idea of the Poincar´e norm, in Proposition 4.10 of [29], it is shown that there exists λ > 1 such that if u, f (u) ∈ M1 ∪ M2, then ρ(f (u))|f⁰(u)| > λρ(u). Let C1 be a positive constant such that ρ(t) > C1 for all t ∈ M₁∪M₂. Then for any u, v ∈ M₁∪M₂ we have |Ru

v ρ(t)dt| > C1|u−v|.

Since c⁰_Mi(u) = 2α⁻¹ρ(u), there exists C₂ > 0 such that |c⁰_Mi(u)| 6 C2 for all u ∈ M_i and i = 1, 2. Therefore, the strong Liapunov condition holds

Q_Mi(f_c(¯u) − f_c(¯v)) − Q_Mj(¯u − ¯v)

In the followings, we list our results of covering relations with the the Liapunov and strong Liapunov conditions.

Theorem 4.30. Let f : R^m → R^m be a C¹ homeomorphism having cover-ing relations with the strong Liapunov condition determined by a transition

matrix A. If g is a C¹ homeomorphism on R^m with |g − f | + kDg − Df k small enough, then there exists a compact subset Λ_g of R^m such that Λ_g is invariant for g and g|Λ_g is topologically conjugate to σ_A.

For small C¹ perturbations of a direct product contracting along the second variable, we have the following result.

Theorem 4.31. Let F (x, y) = (f (x), g(y)) ∈ R^m × R^k be a C¹ homeomor-phism for all x ∈ R^m and y ∈ R^k, where f : R^m → R^m has covering relations with the strong Liapunov condition determined by a transition matrix A, and g : R^k → R^k is a contraction on the closed unit ball B_k such that g(B_k) ⊂ B_k. If G is a C¹ homeomorphism on R^m+k with |G − F | + kDG − DF k small enough, then there exists a compact subset Λ_G of R^m+k such that Λ_G is in-variant for G and G|Λ_G is topologically conjugate to σ_A.

Finally, for a one-parameter family of maps with the singular map F depends only on the phase variable of f , we have the following result.

Theorem 4.32. Let Fλ be a one-parameter family of maps on R^m × R^k satisfying (i) F_λ(x, y) is C¹ continuous as a function jointly of λ ∈ R^`, x ∈ R^m and y ∈ R^k, where λ is a parameter; (ii) F_λ is a homeomorphism on R^m × R^k provided λ 6= 0; and (iii) F0(x, y) = (f (x), g(x)) ∈ R^m × R^k for all x ∈ R^m and y ∈ R^k, where f : R^m → R^m has covering relations with the strong Liapunov condition determined by a transition matrix A, and g : R^m → R^k. Then for each λ sufficiently close to 0, there exists a compact subset Λ_λ of R^m+k such that if λ 6= 0 then Λ_λ is invariant for F_λ and F_λ|Λ_λ is topologically conjugate to σ_A, while Λ₀ is positively invariant for F₀ and F0|Λ0 is topologically semi-conjugate to σ_A⁺.

In order to prove the main results, we need the following proposition which is stated that a homeomorphism having covering relations with the Liapunov condition determined by a transition matrix is topologically conjugate to a two-sided subshift of finite type.

Proposition 4.33. Let f : R^m → R^m be a homeomorphism which has cover-ing relations with the Liapunov condition determined by a transition matrix A. Then there exists a compact subset Λ of R^m such that Λ is maximal in-variant for f in the interior of the union of the h-sets (with respect to A) and f |Λ is topologically conjugate to σ_A.

Proof. We denote by {M_i}^η_i=1 the h-sets with covering relations and the Li-apunov condition for f determined by A as in Definition 4.28, and write s = (. . . , s−1, s₀, s₁, . . .) for s ∈ Σ_A. Define

By using the same argument as in the proof of Proposition 4.25, we have that Λ is a maximal compact invariant set for f in ∪^η_i=1M_i with respect to A and h is a topological semi-conjugacy. Moreover, the covering relations for f on h-sets implies that any boundary point of a h-set can not have a full orbit staying in h-sets. Therefore, Λ is maximal invariant for f in ∪^η_i=1int(M_i) with respect to A.

To prove that h is one to one, we need the following lemma, which is guaranteed by the Liapunov condition.

Lemma 4.34. For any s∈ ΣA, the intersection ∩_n∈Zf⁻ⁿ(Msn) consists of a single point.

Proof. Let s ∈ Σ_A. Then, similar to the proof of Lemma 4.26, we have that the intersection ∩_n∈Zf⁻ⁿ(M_sn) is nonempty. Next, we show the uniqueness of the intersection by contradiction. Assume that u, v ∈ ∩_n∈Zf⁻ⁿ(M_sn) with u 6= v. Since f is a homeomorphism, fⁿ(u) and fⁿ(v) are different points lying in the same h-set M_sn for all n ∈ Z. By the covering relation with the

Liapunov condition, we have that for all n ∈ Z,

QM_sn+1(cM_sn+1◦fⁿ⁺¹(u)−cM_sn+1◦fⁿ⁺¹(v)) > QM_sn(cM_sn◦fⁿ(u)−cM_sn◦fⁿ(v)).

(4.22) That is, the value of Q_Msn at the point c_Msn ◦ fⁿ(u) − c_Msn◦ fⁿ(v) is strictly increasing as n ∈ Z increases. It follows that there exits j ∈ Z such that Q_Msj(c_Msj ◦ f^j(u) − c_Msj ◦ f^j(v)) 6= 0.

First, we consider the case when

Q_Msj(c_Msj ◦ f^j(u) − c_Msj ◦ f^j(v)) > 0. (4.23) By using the compactness of the union ∪^η_i=1M_i, sequentially twice for two sequences, both sequences {f^n+j(u)}^∞_n=0 and {f^n+j(v)}^∞_n=0 have convergent subsequences, say {f^n(i)+j(u)}^∞_i=0 and {f^n(i)+j(v)}^∞_i=0, with the limits, say ¯u and ¯v in ∪^η_i=1M_i, respectively. By the fact that M_i’s are pairwise disjoint and compact, and fⁿ(u), fⁿ(v) ∈ M_sn for all n ∈ Z, there exists α ∈ N such that f^n(i)+j(u), f^n(i)+j(v), ¯u and ¯v are all in the same h-set, namely M_sn(α)+j for all i > α. By the continuity of f , the points f (¯u) and f (¯v) are limits of sequences {f^n(i)+j+1(u)}^∞_i=0 and {f^n(i)+j+1(v)}^∞_i=0, respectively. Again by the same fact as above, there exists a integer β > α such that f^n(i)+j+1(u), f^n(i)+j+1(v), f (¯u) and f (¯v) are all in the same h-set, namely M_sn(β)+j+1 for all i > β. For convenience, we denote N₀ = M_sn(α)+j and N₁ = M_sn(β)+j+1.

By Equation (4.22), we get that for all i > β,

QN0(cN0 ◦ f^n(i)+j(u) − cN0 ◦ f^n(i)+j(v)) > QM_sj(cM_sj ◦ f^j(u) − cM_sj ◦ f^j(v)) By letting i → ∞, it follows from the continuity of Q_N0 and c_N0 that

Q_N0(c_N0(¯u) − c_N0(¯v)) > QM_sj(c_Msj ◦ f^j(u) − c_Msj ◦ f^j(v)).

Thus from Equation (4.23), we have Q_N0(c_N0(¯u) − c_N0(¯v)) > 0 and hence

¯

u 6= ¯v. Since f (¯u), f (¯v) ∈ N1, the Liapunov condition implies that

Q_N1(c_N1 ◦ f (¯u) − c_N1 ◦ f (¯v)) > Q_N0(c_N0(¯u) − c_N0(¯v)). (4.24)

Because that f^n(i)+j+1(u) and f^n(i)+j+1(v) converge to f (¯u) and f (¯v), respec-tively, and both Q_N1 and c_N1 are continuous, we obtain that for some large γ,

QN1(cN1 ◦ f^n(γ)+j+1(u) − cN1 ◦ f^n(γ)+j+1(v)) > QN0(cN0(¯u) − cN0(¯v)).

By using Equation (4.22), we get that for all i > γ + 1,

Q_N0(c_N0◦f^n(i)+j(u)−c_N0◦f^n(i)+j(v)) > Q_N1(c_N1◦f^n(γ)+j+1(u)−c_N1◦f^n(γ)+j+1(v)) Letting i → ∞, it follows from the continuity of Q_N0 and c_N0 that

Q_N0(c_N0(¯u) − c_N0(¯v)) > QN1(c_N1 ◦ f^n(γ)+j+1(u) − c_N1 ◦ f^n(γ)+j+1(v)).

Together with Equation (4.24), this leads to a contradiction.

For the case when Q_Msj(c_Msj◦f^j(u)−c_Msj◦f^j(v)) < 0, by working on the backward orbits of u and v, that is, replacing n and n(i) by −n and −n(i) in the above argument, it leads to a contradiction.

Therefore, the intersection ∩_n∈Zf⁻ⁿ(M_sn) consisting of a single point. We have done the proof of the desired result.

By using Lemma 4.34, we can easily prove that h is one to one. Indeed, let s ∈ Σ_Aand h(z₁) = h(z₂) = s for z₁, z₂ ∈ Λ. Then z₁, z₂ ∈ ∩_n∈Zf⁻ⁿ(M_sn) and hence z₁ = z₂.

Because that the sets Λ and Σ_A are compact and h is a continuous and one to one function, it follows that h is a homeomorphism. This completes the proof of Proposition 4.33.

Now, we begin to prove the main results for with covering relation with the Liapunov condition determined by a transition matrix. In the following, we write A = [aij]_16i,j6η and denote by {Mi}^η_i=1 the pairwise disjoint h-sets with covering relations for f determined by A. For each h-set M_i, let Q_Mi be the quadratic form for the strong cone condition of f.

Proof of Theorem 4.30. Suppose a_ij = 1. Then M_i =⇒ M^f _j holds. By Propo-sition 3.5, if |g − f | is small enough, then M_i =⇒ M^g _j holds. Assume that such a map g is C¹. Before proving that M_i =⇒ M^g _j satisfies the strong Li-apunov condition, let us have some observations. Since M_i =⇒ M^f _j satisfies the strong Liapunov condition, there exists θ_i,j > 0 such that for x, y ∈ M_i,c with x 6= y,

QMj(fc(x) − fc(y)) > QMi(x − y) + θi,j|x − y|². (4.25) For α = i, j, let S_α be the m × m symmetric matrix such that Q_Mα(z) = z^TS_αz for z ∈ R^m. Since f, g and c_Mi are C¹, for x, y ∈ M_i,c, we can define

E_x,y = Z 1

0

Df_c(y + t(x − y))dt and C_x,y = Z 1

0

Dg_c(y + t(x − y))dt.

Then |E_x,y − C_x,y| 6 kDfc− Dg_ck . Since both f_c and g_c are C¹ on the compact set M_i,c, there exists β_i > 0 such that |E_x,y| + |C_x,y| < β_i for all x, y ∈ M_i,c. Thus

|E_x,y^T SjEx,y− C_x,y^T SjCx,y|

6 |Ex,y^T S_jE_x,y− C_x,y^T S_jE_x,y| + |C_x,y^T S_jE_x,y− C_x,y^T S_jC_x,y|

6 βikS_jk kDf_c− Dg_ck . (4.26)

Now we check the strong Liapunov condition for M_i =⇒ M^g _j. Let u, v ∈ M_i,c with u 6= v. By the mean value theorem for integrals, we have that f_c(u) − f_c(v) = E_u,v(u − v) and g_c(u) − g_c(v) = C_u,v(u − v). Thus,

Q_Mj(f_c(u) − f_c(v)) − Q_Mj(g_c(u) − g_c(v))

= (u − v)^T(E_u,v^T S_jE_u,v− C_u,v^T S_jC_u,v)(u − v).

From Equation (4.26), we obtain that

|QMj(fc(u) − fc(v)) − QMj(gc(u) − gc(v))|

6 βikS_jk kDf_c− Dg_ck |u − v|².

Imposing Equation (4.25), we get that Q_Mj(g_c(u) − g_c(v))

> QMj(f_c(u) − f_c(v)) − |Q_Mj(f_c(u) − f_c(v)) − Q_Mj(g_c(u) − g_c(v))|

> QMi(u − v) + θi,j|u − v|²− βikSjk kDfc− Dgck |u − v|²

= Q_Mi(u − v) + (θ_i,j − β_ikS_jk kDf_c− Dg_ck) |u − v|². Finally, we denote

θˆ_i,j = θ_i,j− β_ikS_jk kDf_c− Dg_ck .

Then ˆθ_i,j is independent of u, v ∈ M_i,c. Since c_Mα is C¹ diffeomorphism and M_α is compact for α = i, j, we have that kDf_c− Dg_ck approaches to zero as kDf − Dgk tends to zero. Therefore, if kDf − Dgk is small enough, then θˆ_i,j > 0 and hence M_i =⇒ M^g _j satisfies the strong Liapunov condition.

Since there are at most η² choices of pairs (i, j), from the above, we get that if g is a C¹ continuous map with |g − f | + kDg − Df k small enough, then g has covering relations with the strong Liapunov condition determined by A. In addition, if such maps g are C¹ homeomorphisms, then we have the desired result, by applying Proposition 4.33 and the fact that the strong Liapunov condition implies the Liapunov condition.

Proof of Theorem 4.31. Suppose a_ij = 1. Then the covering relation M_i =⇒^f M_j holds. First, we show that there is a corresponding covering relation for F on h-sets. For α = i, j, let M_α⁰ = M_α× B_k. Then each M_α⁰ is an h-set with c_Mα⁰(x, y) = (c_Mα(x), y), u(M_α⁰) = u(M_α), and s(M_α⁰) = s(M_α) + k. Define a homotopy H : [0, 1] × B_m× B_k→ R^m+k by

H(t, x, y) = (h(t, x), (1 − t)g(y)),

where h is the homotopy for the covering relation M_i =⇒ M^f _j. Then for all x ∈ B_m and y ∈ B_k, we have

H(0, x, y) = (h(0, x), g(y)) = (c_Mj ◦ f ◦ c⁻¹_M

i(x), g(y)) = F_c(x, y), and

H(1, x, y) = (h(1, x), 0).

Thus we have that M_i⁰ =⇒ M^F _j⁰ follows from M_i =⇒ M^f _j.

Next, we show that the strong Liapunov condition is satisfied for M_i⁰ =⇒^F M_j⁰. For α = i, j, define the quadratic form QM_α⁰(x, y) = QMα(x) − |y|². Let cone condition. Since the number of pairs (i, j) is finite, F has covering rela-tions with the strong Liapunov condition determined by A. From Theorem 4.30, the desired result follows.

For i, j ∈ {1, 2, . . . , η} with a_ij = 1, we have that M_i =⇒ M^f _j holds and satisfies the strong Liapunov condition. Thus there exists θ_i,j > 0 such that Q_Mj(f_c(x₁) − f_c(x₂)) > Q_Mi(x₁ − x₂) + θ_i,j|x₁− x₂|² if x₁, x₂ ∈ M_i,c with x₁ 6= x₂. Take a real number ˆθ such that 0 < ˆθ < min{θ_i,j/(1 + L²_i/r²) : i, j ∈ {1, 2, . . . , η}, a_ij = 1}.

Suppose a_ij = 1. For α ∈ {i, j}, let M_α⁰ = M_α × B_k(r). Then each M_α⁰ is an h-set with c_Mα⁰(x, y) = (c_Mα(x), y/r), u(M_α⁰) = u(M_α), and s(M_α⁰) = s(M_α) + k. Define a quadratic form on M_α⁰ by Q_Mα⁰(x, y) = Q_Mα(x) − ˆθ |y|². Then M_i⁰ =⇒ M^F0 _j⁰ holds for a homotopy H : [0, 1] × B_m× B_k → R^m+k defined by

H(t, x, y) = (h(t, x),1 − t r g(c⁻¹_M

i(x))), where h is the homotopy for the covering relation Mi

=⇒ Mf j. Furthermore, we check the strong Liapunov condition. Let (x₁, y₁), (x₂, y₂) ∈ M_i,c⁰ with (x₁, y₁) 6= (x₂, y₂). Then

Therefore, M_i⁰ =⇒ M^F0 _j⁰ satisfies the strong Liapunov condition.

By the finiteness of the pair (i, j), F₀ has covering relations with the strong Liapunov condition determined by A. By Proposition 4.25, there exists a compact subset Λ0 of R^m+k such that Λ0 is positively invariant for F₀ and F₀|Λ₀ is topologically semi-conjugate to σ⁺_A. Since F_λ(x, y) is C¹ in the triple (λ, x, y) of variables, by using the same argument as in the proof of Theorem 4.30, there exists λ0 > 0 such that for all λ with |λ| < λ0, the map F_λ has covering relation with the strong Liapunov condition determined by A. Since F_λ is a homeomorphism on R^m+k provided λ 6= 0, by Proposition 4.33, if 0 < |λ| < λ0 then there exists a compact subset Λλ of R^m+k such that Λ_λ is invariant for F_λ and F_λ|Λ_λ is topologically conjugate to σ_A. We have finished the proof of the theorem.

5 Conclusion

Conclude from this dissertation, we mention some possible future works.

• As the construction of the covering relation, it’s interesting to consider the chaotic dynamics for some nonuniformly hyperbolic systems.

Barreira and Valls [3] consider sequences of Lipschitz maps A_m + f_m such that the linear parts A_m admit a nonuniform exponential di-chotomy, and establish the existence of a unique sequence of topological conjugacies between the maps A_m+ f_m. Also, in [4], they study the re-lation between nonuniform exponential dichotomy and strict Lyapunov sequences. Given such a sequence, they obtain the stable and unstable subspaces from the intersection of images and preimages of the cones defined by each element of the sequence. Use the ideas of nonuni-form exponential dichotomy and strict Lyapunov sequences, we want to construct the covering relations with strong Liapunov condition for the nonuniformly hyperbolic systems.

• It is possible to use the fixed point index theorem to extend the results to the Banach space.

Misiurewicz and Zgliczy´nski [8] use the covering relation in real banach space and the fixed point index theorem to give the result to rigorous estimate topological entropy in case of a one dimensional model (i.e. f is in one dimensional space) where the full system is in infinite dimen-sional real Banach space. As the construction of covering relations in subsection 4.1.2 for map which has a snap-back repeller, we want to extend the result for the compact map which has a snap-back repeller in the real Banach space. Moreover, we want to apply the result to some differential equations.

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