Results

In this subsection, we state our results about the positive topological entropy derived from the topologically crossing homoclinicity. First, we see the result about perturbations of a map.

Theorem 4.14. Let F_λ be a one-parameter family of continuous maps on R^m such that Fλ(x) is continuous as a function jointly of λ ∈ R^` and x ∈ R^m, where λ is a parameter. Assume that F₀(x) = f (x) for all x ∈ R^m, where f : R^m → R^m is a C¹ diffeomorphism with a hyperbolic periodic point which has a topologically crossing homoclinic point. Then there exist an integer

N > 0 and a number λ₀ > 0 such that both f and F_λ with |λ| < λ₀ have topological entropies at least log(2)/N .

Next, if the singular map F₀ depends only on the phase variable of f, we have the following result.

Theorem 4.15. Let F_λ be a one-parameter family of continuous maps on R^m × R^k such that F_λ(x, y) is continuous as a function jointly of λ ∈ R^`, x ∈ R^m and y ∈ R^k, where λ is a parameter. Assume that 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 is a C¹ diffeomorphism with a hyperbolic periodic point which has a topologically crossing homoclinic point, and g : R^m → R^k is a continuous function. Then there exist an integer N > 0 and a number λ₀ > 0 such that both f and F_λ with |λ| < λ₀ have topological entropies at least log(2)/N .

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

Theorem 4.16. Let Fλ be a one-parameter family of continuous maps on R^m × R^k such that F_λ(x, y) is continuous as a function jointly of λ ∈ R^`, x ∈ R^m and y ∈ R^k, where λ is a parameter. Assume that F₀(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 C¹ diffeomorphism with a hyperbolic periodic point which has a topologically crossing homoclinic point, and g : R^m× R^k → R^k is continuous on R^m× S and g(R^m × S) ⊂ int(S) for some compact set S ⊂ R^k homeomorphic to the closed unit ball in R^k. Then there exist an integer N > 0 and a number λ₀ > 0 such that both f and F_λ with |λ| < λ₀ have topological entropies at least log(2)/N .

Denote by p the hyperbolic periodic point of f . Without loss of gen-erality, we may assume that p is a fixed point. Set u = dim W^u(p) and s = dim W^s(p). Since ˆW^u(p) and ˆW^s(p) have a topologically crossing inter-section, we have u + s = m. Let us fix a basis of R^m such that the Jacobian

matrix Df_p of f at p preserves the splitting R^m = R^u⊕ R^s. By the Hartman-Grobman Theorem, there exist a closed neighborhood U of p and a homeo-morphism ϕ of U into R^m such that ϕ(p) = (0, 0) and ϕ(f (z)) = Df_p(ϕ(z)) for z ∈ U . In order to simply our notation, we assume p = (0, 0) and ϕ = id, the identity map on R^m. Thus f is a linear map on U . Write f (x, y) = (L_ux, L_sy) for (x, y) ∈ U, where L_u is a u × u matrix with all eigenvalues greater than one in absolute value and L_s is an s × s matrix with all eigenvalues less than one in absolute value. There exist norms | · |_u and

| · |_s on R^u and R^s, respectively, and constants ρ₁ > 1 and 0 < ρ₂ < 1 such that

|L_ux|_u > ρ1|x|_u and |L_sy|_s 6 ρ2|y|_s for x ∈ R^u and y ∈ R^s. (4.17) Since all norms on R^m are equivalent, we may assume U = B_u × B_s and define the norm | · | on R^m to be the maximum norm of the norms | · |u and

| · |_s on R^u and R^s. Notice that later we still need local coordinates while verifying h-sets in U as required in Definition 3.1. In order to prove the main results we need some lemmas. First, we recall the following lemma in [5]; for readers’ convenience, we repeat their proof below.

Lemma 4.17. [5, Lemma 1.4] Let V be a compact subset of W^u(p) ∩ int(U ).

Suppose we are given positive constants ρ and ε satisfying 0 < ρ < 1 and 0 < ε < d(V, ∂U ). Then for any large enough n ∈ N the following hold:

1. f⁻ⁿ(V ) ⊂ Bu(0, ρ) × {0}; and

2. if (x, 0) ∈ f⁻ⁿ(V ) then fⁿ({x} × B_s) is in U and has diameter less than ε, where the diameter of a bounded set E ⊂ R^m is defined to be sup{|x − y| : x, y ∈ E}.

Proof. Since V ⊂ W^u(p), there exists a positive integer n₁such that f⁻ⁿ¹(V ) ⊂ B_u× {0}. Since f is a C¹ diffeomorphism , we can take a constant K such

that K > sup{kDf_zⁿ¹k : z ∈ U } and ε/(2K) < 1. Let n₂ be an arbitrary positive integer such that

n₂ > max{log(ρ⁻¹)/ log(ρ₁), log(ε/(2K))/ log(ρ₂)}. (4.18) Since f is linear and preserves the splitting R^u× R^s on U , Equations (4.17) and (4.18) imply f⁻ⁿ¹⁻ⁿ²(V ) ⊂ f⁻ⁿ²(B_u × {0}) ⊂ B_u(0, ρ) × {0}. This concludes item 1 of the desired result by considering n = n1+ n2. For item 2, let (x, 0) ∈ f⁻ⁿ¹⁻ⁿ²(V ) and y ∈ B_s. Again Equations (4.17) and (4.18) imply fⁿ²(x, y) = (Lⁿ_u²(x), Lⁿ_s²(y)) ∈ {Lⁿ_u²(x)} × B_s(0, ε/(2K)) ⊂ U . Take any two points in {x} × Bs, say w = (x, y1) and v = (x, y2). Then fⁿ²(w), fⁿ²(v) ∈ U and |fⁿ²(w) − fⁿ²(v)| = |Lⁿ_s²(y₁) − Lⁿ_s²(y₂)| 6 ε/K. By the choice of K, we get that |fⁿ¹⁺ⁿ²(w) − fⁿ¹⁺ⁿ²(v)| < K|fⁿ²(w) − fⁿ²(v)| 6 ε. By considering n = n1+ n2, we have the desired result.

Since the submanifolds ˆW^u(p) and ˆW^s(p) have a topological crossing in-tersection, there exist a point q 6= p and two compact embedded submanifolds V^u of ˆW^u(p) and V^s of ˆW^s(p) such that V^u and V^s form a good pair, and q ∈ V^u∩ V^s. We may assume that both sets V^u and V^s are in int(U ), and V^u has no intersection with the subspace R^u× {0}, based on the following lemma.

Lemma 4.18. For any sufficiently integer n ∈ N, there exist submanifolds V_n^u of ˆW^u(p) and V_n^u of ˆW^s(p) such that V_n^u and V_n^s form a good pair with the same oriented intersection number as good pair V^u and V^s, V_n^u ⊂ fⁿ(V^u), V_n^s ⊂ fⁿ(V^s), and V_n^u∪ V_n^s ⊂ int(U ).

Proof. First, we show that fⁿ(V^u) and fⁿ(V^s) form a good pair for n ∈ N.

Since f is a C¹ diffeomorphism, fⁿ(V^u) and fⁿ(V^s) are compact embedded C¹ submanifolds of ˆW^u(p) and ˆW^s(p), respectively. Since V^u ⊂ ˆW^u(p), V^s ⊂ ˆW^s(p), and ∂V^u∩V^s= V^u∩∂V^s = ∅, we also have ∂fⁿ(V^u)∩fⁿ(V^s) = fⁿ(V^u) ∩ ∂fⁿ(V^s) = ∅.

Let δ be a constant such that

0 < δ < min{d(∂fⁿ(V^u), fⁿ(V^s)), d(fⁿ(V^u), ∂fⁿ(V^s))}.

Since fⁿis continuous on the compact set V^u∪V^s, there exists a constant ε such that 0 < ε < min{d(∂V^u, V^s), d(V^u, ∂V^s)} and if x, y ∈ V^u ∪ V^s with |x − y| < ε then |fⁿ(x) − fⁿ(y)| < δ. Since V^u and V^s form a good pair, for such an ε, there exists a homotopy h₀ satisfying item (2a)-(2d) of Definition 4.12. Define a homotopy hn = fⁿ◦ h0 ◦ f⁻ⁿ. It is obviously true that h_n(0, ·) = id and h_n(1, ·) is an embedding. By item (2b) of Definition 4.12, for z ∈ fⁿ(V^u) ∪ fⁿ(V^s) and t ∈ [0, 1], we have

|h_n(t, z) − z| = |fⁿ(h₀(t, f⁻ⁿ(z))) − fⁿ(f⁻ⁿ(z))| < δ.

Moreover, h_n(1, fⁿ(V^u)) and fⁿ(V^s) are transverse submanifolds and the oriented intersection number I(h_n(1, fⁿ(V^u)), fⁿ(V^s)) = I(h₀(1, V^u), V^s) is nonzero. Thus, fⁿ(V^u) ⊂ ˆW^u(p) and fⁿ(V^s) ⊂ ˆW^s(p) form a good pair.

If fⁿ(V^u) ∪ fⁿ(V^s) ⊂ int(U ), then we are done by taking V_n^u = fⁿ(V^u) and V_n^s = fⁿ(V^s). Otherwise, since p is a hyperbolic fixed point with topologically crossing homoclinic point(s) in V^u ∩ V^s which has nonzero oriented intersection number, by letting n large enough, there exists q ∈ V^u ∩ V^s such that if we denote by V_n^u and V_n^s the connected components of fⁿ(V^u) ∩ (B_u(0, 4/5)) × (B_s(0, 4/5)) and fⁿ(V^s) ∩ (B_u(0, 4/5)) × (B_s(0, 4/5)) containing the point fⁿ(q), respectively, ∂V_n^u ∩ V_n^s = V_n^u ∩ ∂V_n^s = ∅ and I(h_n(1, V_n^u), V_n^s) = I(h_n(1, fⁿ(V^u)); fⁿ(V^s)). Repeating the above argument, we have that V_n^u ⊂ ˆW^u(p) and V_n^s⊂ ˆW^s(p) form a good pair with the same oriented intersection number as the good pair V^u and V^s. We have finished the proof of the desired result.

Set V₂ = V^u. Since π_u(q) = 0, there is a constant η such that 0 < η < 1 and B_u(0, η) ⊂ π_u(V₂)\π_u(∂V₂). Denote V₁ = B_u(0, η) × {0}.

We shall construct two disjoint h-sets. Let ρ be a constant such that 0 < ρ < η. Denote R = B_u(0, ρ) × B_s. Let ε > 0 be so small that the closed neighborhoods of V₁ and V₂ are disjoint and contain in int(U ) and that the closed ε-neighborhoods of ∂V^u and ∂B_u(0, η) × {0} are contained in int(U ) \ R. Then

ε < min{d(V₁, ∂U ), d(V₂, ∂U )}.

By applying Lemma 4.17 to V1 and V2, we can pick a common integer N such that f^−N(V₁) ∪ f^−N(V₂) ⊂ B_u(0, ρ) × {0} and if (x, 0) ∈ f^−N(V₁) ∪ f^−N(V₂) then f^N({x} × B_s) is in U and has diameter less than ε. Write f^−N(q) = (q0, 0). Since f is a diffeomorphism, f^−N(V1) and f^−N(V2) are disjoint. More-over, since f is C¹, V₂ is a C¹ submanifold of ˆW^u(p) and hence there exists a C¹ diffeomorphism ζ from R^u to R^u such that ζ(π_u(f^−N(ψ(x)))) = x for all x ∈ Bu, where ψ is a C¹ parametrization of V2 on Bu such that V2 = ψ(Bu) and ψ(0) = q (mentioned in Lemma 4.13). Since f (x, y) = (L_ux, L_sy) for (x, y) ∈ U under the Hartman-Grobman linearization setting at the begin-ning of this subsection, we have f^−N(V1) = L^−N_u (Bu(0, η)) × {0}. Define M₁ = π_u(f^−N(V₁)) × B_s and M₂ = π_u(f^−N(V₂)) × B_s; or equivalently define M₁ = L^−N_u (B_u(0, η)) × B_s and M₂ = π_u(f^−N(ψ(B_u))) × B_s. Then M₁ and M2 are disjoint h-sets with u(M1) = u(M2) = u, s(M1) = s(M2) = s, and c_M1(x, y) = (L^N_ux/η, y) and c_M2(x, y) = (ζ(x), y) for all (x, y) ∈ R^u× R^s.

Next, we show that there are covering relations among M₁ and M₂. Lemma 4.19. The following covering relations hold:

M_i ^f

N

=⇒ M_j for i, j ∈ {1, 2}.

Proof. Define a homotopy H on R^m from f^N(x, y) to πu◦ f^N(x, 0) by H(t, x, y) = (1 − t)f^N(x, y) + t(π_u(f^N(x, 0)), 0),

for (x, y) ∈ R^u× R^s and t ∈ [0, 1]. For i, j ∈ {1, 2}, we set a homotopy h^j_i induced from H by h^j_i(t, x, y) = c_Mj(H(t, c⁻¹_M

i(x, y))) for (x, y) ∈ R^u× R^s and

t ∈ [0, 1], and define A^j_i(x) = π_u(h^j_i(1, x, 0)) for x ∈ R^u. Since h^j_i(1, x, y) is independent of y and lies on the subspace R^u× {0}, we get that h^j_i(1, x, y) = (A^j_i(x), 0) for x ∈ R^u. Moreover, by the choice of N , we have

H(t, M_i⁻) ∩ Mj = ∅ and H(t, Mi) ∩ M_j⁺= ∅ for t ∈ [0, 1].

It follows that Condition 1 and 2 of Definition 3.2 are satisfied with h = h^j_i and ϕ = A^j_i .

For Condition 3 of Definition 3.2, we first show that deg(A^j_i, Bu, 0) 6= 0 for i = 1 and j ∈ {1, 2}. By the definition of homeomorphisms c_Mi, we get that f^N ◦ c⁻¹_M

1(U ) ⊂ U and f^N ◦ c⁻¹_M

2(U ) ⊂ U . Hence on B_u, the map A¹₁ is linear and the map A²₁ is C¹, in fact, they are of the following forms: for x ∈ B_u,

A¹₁(x) = π_u(h¹₁(1, x, 0)) = L^N_ux, A²₁(x) = π_u(h²₁(1, x, 0)) = ζ(ηx).

Since L_u is a u × u matrix with all eigenvalues greater than one in absolute value, by item 8 of the properties of local Brouwer degree listed in subsection 3.2, we get

deg(A¹₁, B_u, 0) = sgn(det(L^N_u)) 6= 0.

The choice of N implies π_u(f^−N(V₂)) ⊂ B_u(0, ρ) ⊂ B_u(0, η) and hence the equation ζ(x) = 0 has a unique solution, namely q0, and q0 ∈ Bu(0, η). Since π_u(f^−N(V₂)) is a u-dimensional C¹ submanifold, 0 is a regular value for ζ and sgn(det Dζ_q0) 6= 0. It follows from items 8 again and Proposition 3.6 that

deg(A²₁, B_u, 0) = sgn(det Dζ_q0) · 1 6= 0.

Next, we shall show that deg(A^j_i, B_u, 0) 6= 0 for i = 2 and j ∈ {1, 2}

as applications of Lemma 4.13. By the definitions of the homeomorphisms

c_Miand the linearization of f on U , we get that for x ∈ B_u, A¹₂(x) = ϕ ◦ π_u◦ ψ(x),

A²₂(x) = ζ ◦ π_u◦ ψ(x),

where ϕ is a map on R^u defined by ϕ(x) = L^N_ux/η. By the choice of η, there exists a bounded connected component, namely ∆, of R^u\ π_u(ψ(∂B_u) such that 0 = ϕ⁻¹(0) ∈ ∆. Since ϕ is linear, Proposition 3.6 implies that

deg(A¹₂, B_u, 0) = deg(ϕ ◦ π_u◦ ψ, B_u, 0) = sgn(det(L^N_u/η)) deg(π_u ◦ ψ, B_u, 0) Note that ψ is a parametrization of V₂. Since V₂ and V^s form a good pair with the oriented orientation number not zero, by Lemma 4.13, we have deg(πu◦ ψ, Bu, 0) 6= 0. Therefore, deg(A¹₂, Bu, 0) 6= 0.

Similarly, by the choices of η and N , we get ζ⁻¹(0) = q₀ ∈ ∆. Since ζ is C¹, Proposition 3.6 gives us that

deg(A²₂, B_u, 0) = deg(ζ ◦ π_u◦ ψ, B_u, 0) = sgn(det Dζ_q0) deg(π_u◦ ψ, B_u, 0).

It follows that deg(A²₂, B_u, 0) 6= 0.

We have finished the proof of the desired result.

Finally, we are in position to prove our theorems.

Proof of Theorem 4.14. By applying Lemma 4.19 and Proposition 3.5, there exists λ₀ > 0 such that if |λ| < λ₀ then

M_i ^F

N

=⇒ Mλ _j for for i, j ∈ {1, 2}.

Let θ be a positive integer and |λ| < λ₀. Consider any closed loop M_i0 ^F

N

=⇒ Mλ _i1 ^F N

=⇒ · · ·λ ^F N

=⇒ Mλ _iθ,

with each i_α ∈ {1, 2} and i_θ = i₀. By using Theorem 3.3, F_λ^N has a periodic point x = x(λ) ∈ int(M_i0) such that F_λ^{N θ}(x) = x. Since there are 2^θ choices

of such closed loops, F_λ^N has at least 2^θ periodic points in M₁ ∪ M₂. These periodic points provide a (θ, δ)-separated set for F_λ^N as long as δ is a positive number less than the distance of M₁ and M₂. Since θ is arbitrarily chosen, we have h_top(F_λ^N) > log(2) and so htop(F_λ) > log(2)/N > 0.

Lemma 4.20. The following covering relations hold:

M_i^{0 F}

N

=⇒ M0 _j⁰ for i, j ∈ {1, 2}.

Proof. Let i, j ∈ {1, 2} be arbitrary. We define a homotopy ˆh^j_i(t, x, y, z) = (h^j_i(t, x, y),1 − t

r g ◦ f^{N −1}(c⁻¹_M

i(x, y))), where h^j_i is the homotopy for the covering relation M_i ^f

N B_k, Condition 1 and 2 of Definition 3.2 are satisfied follows from the anal-ogous properties for h^j_i stated in the proof of Theorem 4.14. Finally, notice that

ˆh^j_i(1, x, y, z) = (h^j_i(1, x, y), 0).

Therefore, Condition 3 of Definition 3.2 is satisfied.

By applying Lemma 4.20 and Proposition 3.5, there exists λ₀ > 0 such that if |λ| < λ₀ then the following covering relations hold for F_λ^N:

M_i^{0 F}

N

=⇒ Mλ _j⁰ for i, j ∈ {1, 2}. (4.19) As in the proof of Theorem 4.14, the covering relations listed in Equation (4.19) implies the h_top(F_λ) > log(2)/N > 0 with |λ| < λ0.

Proof of Theorem 4.16. Define G_λ = (id, c)◦F_λ◦(id, c)⁻¹, where c is a home-omorphism from S to B_k. Then the topological entropies of G_λ and F_λ are equal. By applying the above argument as in the proof of Theorem 4.15 to the family G_λ while the corresponding c_M of a covering relation N =⇒ M is^Gλ the identity map, we have the desired result.