國立交通大學應用數學系
博士論文
具覆蓋關係動態函數之高維度擾動的拓樸混沌
Topologically chaos for multidimensional
perturbations of maps with covering relations
研究生: 呂明杰
指導教授: 李明佳教授
中華民國一百零一年七月
具覆蓋關係動態函數之高維度擾動的拓樸混沌
Topologically chaos for multidimensional
perturbations of maps with covering relations
研究生:呂明杰 Student: Ming-Jiea Lyu
指導教授: 李明佳 Advisor: Ming-Chai Li
國立交通大學應用數學系
博士論文
A Thesis
Submitted to Department of Applied Mathematics
National Chiao Tung University
in partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
in
Applied Mathematics
July 2012
具覆蓋關係動態函數之高維度擾動的拓樸混沌
研究生:呂明杰 指導教授:李明佳 教授
國立交通大學
應用數學系
摘要
本論文主要研究高維度系統的拓樸動態，其中系統是擾動
由 F(x,y)=(f(x),g(x,y))形式之系統且滿足低維度函數 f 為
一連續函數。首先我們會證明如果當低維度函數 f 具有返回
擴張固定點，其微小的 C
1擾動同樣具有返回擴張固定點。
假設函數g具有局部抑制的區域且系統沿著一連續的參數
群{F
λ}滿足F
0=F。我們會證明如果當低維度函數f為一維度函
數且具有正的拓樸熵或f為一高維度函數具有返回擴張固定
點，則對於所有夠小的參數λ，F
λ也會具有正的拓樸熵。並
且 我 們 證 明 如 果 當 f 為 一 微 分 同 胚 具 有 topologically
crossing homoclinic point時，則對於參數λ夠接近0時，
更進一步地，我們證明當f具有由轉移矩陣A決定的覆蓋關
係時，則F的任意微小C
0擾動系統會存在一緊緻正向的不變集
且當系統限定在此不變集上時會拓樸半共軛到由A生成的單
邊有限型子轉移。此外，如果覆蓋關係滿足strong Liapunov
condition且函數g為一壓縮函數，則我們會證明出F的任意
微小C
1擾動同胚會存在一緊緻的不變集且當系統限定在此不
變集上時會拓樸共軛到由A生成的雙邊有限型子轉移。
Topologically chaos for multidimensional
perturbations of maps with covering relations
Student: Ming-Jiea Lyu Advisor: Ming-Chai Li
Department of Applied Mathematics
National Chiao Tung University
Abstract
In this dissertation, we investigate topological dynamics of high-dimensional systems which are perturbed from a continuous map f of the following form F(x,y) = (f(x),g(x,y)). First, we show that if the lower dimensional map f has a snap-back repeller, then the small C1 perturbation of f also has a snap-back repeller.
Assume that g is locally trapping and the system is along a one-parameter continuous family {Fλ} such that F0 = F. We show that if f is a one dimensional map
and has positive entropy, or f is a high-dimensional map and has a snap-back repeller then {Fλ} has a positive topological entropy for all small parameter λ. Also, we
show that if f is a C1 diffeomorphism having a topologically crossing homoclinic point, then {Fλ} has positive topological entropy for allλclose enough to 0.
Moreover, we show that if f has covering relations determined by a transition matrix A, then any small C0 perturbed system of F has a compact positively invariant set restricted to which the perturbated system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C1
system is topologically conjugate to the two-sided subshift of finite type induced by A.
誌 謝
在交大讀博士班六年的時光中，充滿了許許多多的喜怒哀樂。雖然中途在準備 英文考試方面有一些挫折，不過最後還是通過考試進而完成學業。首先，感謝指 導教授李明佳老師在學業上給了我一些指導與建議，這對我在動態系統的概念上 有很大的幫助。由於他的包容與支持，讓我能在迷途中，有了一盞明燈指引方向; 從他生活處事與對數學研究態度上，我看到了一個數學家的嚴謹與堅持。另外， 他也常常鼓勵我參加國內外的研討會，讓我能增廣見聞及培養我的國際觀，藉以 提升數學深度。 其次，感謝口試委員莊重教授、石至文教授、陳國璋教授、夏俊雄教授的寶貴 建議。莊重老師在口試時給了我很多建議與想法，對於我研究的問題上希望能有 更進一步的想法與突破。另外，感謝石至文與陳國璋老師邀請我參加研討會與活 動，讓我能有機會上台去演講且有機會多與其他的學者交流，這是非常難得的經 驗。最後，感謝夏俊雄老師在我未來的研究計畫給與我一些建議。 在交大念書的這一段時間中，十分感謝教導過我的老師們。吳培元老師的實變 分析課程讓我在實變分析與泛含分析上受益良多。指導教授李明佳老師與林琦焜 老師的微分方程專題讓我們能一步一步的了解動態系統與微分方程間的各種層 面與現象。另外，同學之間讀完論文一起提出問題討論的上課方式也讓我們得到 很多啟發。 在博士班的生活中，感謝各位學長姐、同學、學弟妹的關心與支持。其中特別 感謝同研究室的張志鴻學長、陳耀漢、龔柏任與黃俊銘及隔壁研究室的胡文貴、 吳恭儉、蔡明耀、李信儀、陳德軒等，在與他們的相處與討論中，常常能充滿歡 樂，同時也讓我獲益良多。 感謝我的父母與家人。因為有他們的支持，才能有今天的我。無論遇到甚麼困 難，他們始終支持我，這種無形的助力對我而言是非常巨大的幫助。雖然最後兩 年家裡發生很多事情，令人感傷，不過最後還是能完成學業。 最後，感謝大家一路上的陪伴與幫助，我才能順利地從博士班畢業，今後我會 繼續努力地朝目標邁進。在此，由衷地感謝大家，謝謝你們。 呂明杰 101/07/23 於交大Contents
1 Introduction 1 2 Snap-back repeller 6
2.1 Preliminaries . . . 7
2.2 Persistence of snap-back repeller . . . 9
3 Covering relations 14 3.1 Background and applications . . . 14
3.2 Properties of local Brouwer degree . . . 17
4 Topological dynamics for multidimensional perturbations 19 4.1 Snap-back repellers and one dimensional maps . . . 19
4.1.1 One dimensional maps . . . 19
4.1.2 Higher dimensional maps . . . 22
4.2 Topologically crossing homoclinicity . . . 32
4.2.1 Background . . . 32
4.2.2 Results . . . 34
4.3 Liapunov condition . . . 43
4.3.1 Covering relations determined by a transition matrix . 43 4.3.2 Liapunov and strong Liapunov condition . . . 48 5 Conclusion 59 References 60
1
Introduction
In this dissertation, we mainly study the perturbation from a map f on the lower dimensional phase space, which has some dynamical properties (positive topological entropy, snap-back repeller, topologically crossing ho-moclinicity, covering relations determined by a transition matrix, etc.) to continuous map G on a high dimensional space such that G is a small per-turbation of the singular map F which is one of the following cases:
(i) F (x) = f (x) ∈ Rm;
(ii) F (x, y) = (f (x), g(x)) ∈ Rm× Rn;
(iii) F (x, y) = (f (x), g(x, y)) ∈ Rm× Rn and g(Rm× S) ⊂ int(S) for some
compact set S ⊂ Rn homeomorphic to the closed unit ball in Rn, where
int(S) denote the interior of S;
(iv) F (x, y) = (f (x), g(y)) ∈ Rm × Rn, where g is a contraction on the
closed unit ball in Rn and has the unique fixed point in the interior of the unit ball.
The question we discussed is the following.
(#) Does G have chaotic dynamics?
The map G in cases (ii)-(iv) is considered as multidimensional perturba-tion of f due to bigger dimension of phase space, while G in case (i) is a usual perturbation of f and they have the same phase space. The singular map F in cases (ii)-(iv) can be considered as the skew product (f (x), q(x, y)) with different strength on trapping region of q(x, y): vertical contraction q(x, y) = g(x) for case (ii), locally trapping q(Rm× S) ⊂ int(S) for case (iii),
and horizontal contraction for q(x, y) = g(y).
In 1975, Li and Yorke [18] introduced the mathematical definition of chaos and established a very simple criterion: “period three implies chaos” for its
existence in the real number. This criterion played a key role in predicting and analyzing one-dimensional chaotic dynamical systems. In 1978, Marotto [19] wanted to study chaos for higher dimensional discrete dynamical systems and he proved that “if a differentiable map has a snap-back repeller then it exhibits the sense of Li-Yorke chaos”. Based on Marotto’s argument, Blanco Garc´ıa [2] showed that a snap-back repeller implied positive topological en-tropy. Here, in Section 2, we give a definition of snap-back repeller slightly different from Marotto’s in [19, 23] so that it is independent of norms and the mentioned results of Marotto and Blanco Garc´ıa still hold obviously. Also we use the implicit function theorem in Banach spaces to prove that any small C1 perturbation of a (possibly noninvertible) system with a snap-back
repeller has a snap-back repeller and exhibits chaos. This establishes one kind of result addressing question (#) in case (i) for snap-back repeller, refer to [13].
In Section 3-4, we focus on the results about topological entropy which is a quantitative measurement of how chaotic a map is. In fact, it is determined by how many different orbits there are for a given map. The methodology we used to study the question (#) is based on the concept of covering re-lation which was introduced by Zgliczy´nski in [33, 34], see Section 3 for its background and applications. It allows one to prove the existence of pe-riodic points, the symbolic dynamics and the positive topological entropy without using hyperbolicity. Also, the persistence of covering relation under small perturbation allow one to consider the multidimensional perturbation of systems.
There are several existing literature investigating the question (#) about topological entropy. For the case when f is an interval map and g = 0 in a real Banach space, Misiurewicz and Zgliczy´nski in [8] proved that lim infλ→0htop(Fλ) > htop(f ). For the planar case (ie. m = n = 1),
Fλ(x, y) = (ϕ(x, λy), x) and λ ∈ R and the other one that is Fλ(x, y) =
(ϕ(x, λ1y), g(λ2x, y)), λ = (λ1, λ2) ∈ R2, and the map y 7→ g(0, y) has a
stable fixed point. Assuming the map x 7→ ϕ(x, 0) is C1 and has a
snap-back repeller, he showed that for all λ near 0, the map Fλ has a transverse
homoclinic point. His method relies heavily on the planar structure of the map F0 and the Birkhoff-Smale transverse homoclinic point theorem. Also,
the results from [11, 17] about difference equations can be applied to ques-tion (#) for the topological entropy, but these are in fact perturbaques-tions of one-dimensional maps.
In subsection 4.1, we establish two kinds of results addressing question (#) in cases (ii) and (iii) for f having positive topological entropy in one dimensional space and snap-back repeller in higher dimensional space, along a one-parameter continuous family {Fλ} such that F = F0 and G = Fλ with
small parameter λ. First we show that if f is a one-dimensional map (without any additional assumption) then lim infλ→0htop(Fλ) > htop(f ) (see Theorem
4.1 and 4.2). Second, we allow f to be possibly high-dimensional map and show that if f has a snap-back repeller then htop(Fλ) > 0 for all λ near enough
0 (see Theorems 4.9 and 4.10), refer to [16]. Moreover, as a by-product of using covering relation, we give a new proof of Blanco Garcia’s result in [2] that the existence of a snap-back repeller implies positive topological entropy (see Proposition 4.8).
Theae results are applicable to a high-dimensional version of the H´ enon-like maps. Define a family of maps Hb(x, y) on Rm × Rn, with parameter
b ∈ R`, by its components, for x = (x
1, ..., xm) and y = (y1, ..., yn), ¯ xi = ai− x2i + oi(b)ϕi(x, y), 1 6 i 6 m, ¯ yj = gj(x, y), 1 6 j 6 n,
where each ai is a constant, oi, ϕi, gj are real-valued continuous functions
and limb→0oi(b)/|b| = 0. If m = n = 1, one can reduce Hb to the original
well as from [11, 17, 21]. For the general case when m > 1 and n > 1, we assume that each gj is either dependent only on x or bounded (hence, the
conditions in form (ii) or (iii) are satisfied, respectively). At the singular value b = 0, the first m components of H0, i.e. x¯i = ai − x2i for 1 6
i 6 m, form a decoupled map from Rm into itself, and such a map has
a positive topological entropy or a snap-back repeller by choosing suitable ai. By applying the results about topological entropy of multidimensional
perturbations with snap-back repellers on lower dimensional map, we get that htop(Hb) > 0 for all b sufficiently near 0.
The idea of a topologically crossing intersection of two submanifolds is from [5, 7, 8] (see subsection 4.2.1 for background). The methodology we use to study the question (#) with f having topologically crossing intersection is based on the construction of topological horseshoe, given by Burns and Weiss in [5], and the concept of covering relations. Topologically crossing homo-clinicity guarantees existence of covering relations on which f has both topo-logical contraction and expansion directions. Unlike the discuss in subsection 4.1, the covering relations have only expansion direction for an interval map f with positive topological entropy or a map f with a snap-back repeller. In subsection 4.2.2, we establish the results addressing question (#) in cases (i)-(iii) for f being a C1 diffeomorphism with a hyperbolic periodic point
which has a topologically crossing homoclinic point, along a one-parameter continuous family {Fλ}. We show that Fλ has positive topological entropy
for all λ close to 0, refer to [14].
In subsection 4.3.1, we assume that f has covering relations determined by a transition matrix A (see Definition 4.21) and show that for cases (i)-(iii), if G is C0 close to F , then G has an isolated invariant set to which the
re-striction G is topologically semi-conjugate to the one-sided subshift of finite type, denote by σA+, and hence the topological entropy of G is greater than the logarithm of the spectral radius of A (see Theorems 4.22-4.24). In
addi-tion, in subsection 4.3.2, if the covering relations satisfy the strong Liapunov condition (see Definition 4.28), then we conclude that if a homeomorphism G is C1 close to F , then G has an isolated invariant set to which the restriction
of G is topologically conjugate to the two-side subshift of finite type, denote by σA, for the cases (i) and (iv) provided that F is a homeomorphism (see
Theorems 4.30 and 4.31), and for the case (ii) provided that G is perturbed from F along a one-parameter continuous family {Fλ} such that F = F0 and
G = Fλ with small |λ| 6= 0 (see Theorem 4.32), refer to [15].
In particular, one can apply the last result to the H´enon-like like family Fλ(x, y) = (f (x) + p(λ, x, y), q(λ, x, y)), where f is the logistic map f (x) =
µx(1 − x) with µ > 4, p and q are C1 continuous functions of (λ, x, y) such
that Fλ is a homeomorphism for λ = 0, and h(0, x, y) = 0 for all (x, y) and
q(0, x, y1) = q(0, x, y2) for all x, y1 and y2. The map f has covering relations
which are determined by the 2 × 2 matrix with all entries one and satisfy the strong Liapunov condition (see Example 4.29). Thus for sufficiently small |λ| 6= 0, the map Fλ has an isolated invariant set on which Fλ is topologically
conjugate to the 2-shift. By setting p(λ, x, y) = λy and q(λ, x, y) = x, the family Fλ becomes the original H´enon family.
2
Snap-back repeller
In this section, we study the snap-back repellers. Recently, Marotto [23] redefined back repeller and stated that his early result in [19]: “a snap-back repeller implies Li-Yorke chaos” is still correct. First, in here, we list the Marotto’s definition of snap-back repeller in [23].
Definition 2.1 ([23], Definition 1). Suppose z is a fixed point of a differ-entiable map f with all eigenvalues of Df (z) exceeding 1 in magnitude, and suppose there exists a point x0 6= z in a repelling neighborhood of z, such that
xM = z and det(Df (xk)) 6= 0 for 1 6 k 6 M , where xk = fk(x0). Then z is
called a snap-back repeller of f .
Marotto’s definition depend on the norms of the phase space. Now we give our definition of a snap-back repeller which is slightly different form Marotto’s definition. It is independent of norms.
Definition 2.2. Let f : Rk → Rk be a differentiable function. A fixed point
w0 for f is called a snap-back repeller if (i) all eigenvalues of Df (w0) are
greater than one in absolute value and (ii) there exists a sequence {w−n}n∈N
such that w−1 6= w0, limn→∞w−n = w0, and for all n ∈ N, f (w−n) = w−n+1
and det(Df (w−n)) 6= 0.
Based on Marotto’s argument, Blanco Garc´ıa [2] showed a snap-back re-peller implies positive topological entropy. The mentioned results of Marotto and Blanco Garc´ıa under our definition till hold. Roughly speaking, a snap-back repeller of a map is a repelling fixed point associated with which there is a transverse homoclinic point. Notice that if there exists a norm | · |∗ on Rk
such that for some constants δ > 0 and λ > 1, one has that |f (x) − f (y)|∗ >
λ|x − y|∗ for all (x, y) ∈ B(w0, δ) where B(w0, δ) = {x ∈ Rk : |x − w0|∗ < δ},
then f is one-to-one on B(w0, δ) and f (B(w0, δ)) ⊃ B(w0, δ); hence item (ii)
fm(q) = w0 and det(Dfm(q)) 6= 0 for some positive m. In fact, item (i)
implies that such a norm must exist (refer to [29, Theorem V.6.1]). Further-more, if all eigenvalues of (Df (w0))TDf (w0) are greater than one, then such
a norm can be chosen to be the Euclidean norm on Rk (see [12, Lemma 5]).
2.1
Preliminaries
In this subsection, we recalled the result about “a snap backer repeller im-plies Li-Yorke Chaos” which was proved by Marotto in [19] and [23] and “a snap-back repeller implies positive topological entropy” which was proved by Blanco Garc´ıa in [2].
First, we describe the mathematical sense of chaos introduced by Li and Yorke in [18]:
Theorem 2.3 ([18], Theorem 1). Let J be an interval in R and let F : J → J be continuous. Assume that there is a point a ∈ J , for which the points b = F (a), c = F2(a) and d = F3(a), satisfy
d 6 a < b < c ( or d > a > b > c). Then:
1. for every k ∈ 1, 2, . . ., there is a periodic point in J having period k; 2. there is an uncountable set S ⊂ J (containing no periodic points), which
satisfies the following conditions: (a) for every p, q ∈ S with p 6= q,
lim sup n→∞ |Fn(p) − Fn(q)| > 0 and lim inf n→∞ |F n (p) − Fn(q)| = 0;
(b) for every p ∈ S and periodic points q ∈ J , lim sup
n→∞
|Fn(p) − Fn(q)| > 0.
In [19], Marotto studied the Li-Yorke theorem to higher dimensional dis-crete dynamical systems.
Theorem 2.4 ([19], Theorem 1). Let f : Rk → Rk possess a snap-back
repeller. Then f exhibits Li-Yorke chaos, that is, there exist
1. a positive integer N such that if m > N is an integer, the map f has a point of period m;
2. an uncountable set S containing no periodic points of f such that (a) if x, y ∈ S with x 6= y, then
lim sup
n→∞
|fn(x) − fn(y)| > 0; (b) if x ∈ S and y is a periodic point for f , then
lim sup
n→∞
|fn(x) − fn(y)| > 0; (c) f (S) ⊂ S; and
3. an uncountable subset S0 of S such that if x, y ∈ S0, then
lim inf
n→∞ |f
n(x) − fn(y)| = 0.
Next, we review the background of topological entropy. Let (X, d) be a compact metric space and let f : X → X be a continuous map. For n ∈ N, the function
dn,f(x, y) = max 06k<nd(f
k(x), fk(y))
measures the maximum distance between the first n iterates of x and y. For n ∈ N and > 0, a set S ⊂ X is called (n, )-separated for f provided
dn,f(x, y) > for every pair of points x, y ∈ S with x 6= y. The number of
different orbits of length n (as measured by ) is defined by
r(n, , f ) = max{#(S) : S ⊂ X is a (n, )-separated set for f },
where #(S) is the number (cardinality) of elements in S. In order to measure the growth rate of r(n, , f ) as n increases, we define
h(, f ) = lim sup
n→∞
log(r(n, , f )) n .
Finally, we consider h(, f ) varies as goes to 0 and define the topological entropy of f as
htop(f ) = lim
→0+h(, f ).
Moreover, let f : X → X be a continuous function where X is a metric space. Here, the topological entropy of f is defined to be the supremum of topological entropies of f restricted to compact invariant sets. Refer to [29] for more background.
Blanco Garc´ıa [2] proved that a snap-back repeller implies positive topo-logical entropy.
Theorem 2.5 ([2], Theorem 1). Let F : Rk → Rk be a differentiable map.
If F has a snap-back repeller, then F has positive topological entropy.
2.2
Persistence of snap-back repeller
In this subsection, we show the persistence of snap-back repeller for small C1 perturbations by using the implicit function theorem in Banach spaces (refer to Lang’s textbook [31, Theorem 6.2.1]). Let k be a positive integer, | · |2 be
the Euclidean norm on Rk, and || · ||2 be the operator-norm on the space of
linear maps on Rk induced by | · | 2.
Theorem 2.6. Let f : Rk → Rk be a C1 map on Rk with a snap-back
repeller. If g is a C1 map on Rk such that |f − g|
enough, then g has a snap-back repeller, exhibits Li-Yorke chaos, and has positive entropy.
Proof. Let x0 be a snap-back repeller of f and {x−n}n∈N be its
correspond-ing homoclinic orbit with x−1 6= x0, limn→∞x−n = x0, and for all n ∈ N,
f (x−n) = x−n+1 and det(Df (x−n)) 6= 0. Since x0 is a fixed point of f
and all eigenvalues Df (x0) are greater than one in absolute value, there
ex-ists a norm | · |∗ on Rk such that for some constants δ0 > 0 and λ0 > 1,
one has that |f (x) − f (y)|∗ > λ0|x − y|∗ for all x, y ∈ B(x0, δ0), where
B(x0, δ0) = {x ∈ Rk : |x − x0|∗ < δ0}. Thus f is one-to-one on B(x0, δ0)
and f (B(x0, δ0)) ⊃ B(x0, δ0). Let || · ||∗ denote the operator-norm in the
space of linear maps on Rk induced by | · |
∗. Let λ1 be a constant with
1 < λ1 < λ0 and let U (f, λ0− λ1) denote the set of all C1 maps g on Rk with
|f − g|∗ + ||Df − Dg||∗ < λ0 − λ1. Then for any g ∈ U (f, λ0 − λ1) and x,
y ∈ B(x0, δ0), we have that
|g(x) − g(y)|∗ > |f (x) − f (y)|∗− |(g − f )(x) − (g − f )(y)|∗ (2.1)
> [λ0− (λ0 − λ1)]|x − y|∗ = λ1|x − y|∗;
hence, g is one-to-one on B(x0, δ0). Let δ > δ0 be a constant so that
{x−n}n∈N ⊂ B(x0, δ0). Denote by W the closure of B(x0, δ0). Then W
is a compact subset of Rk. Let S be the space of C1 functions from W
to Rk endowed with the usual C1 topology d
C1 which is induced from the
norm | · |∗ on Rk. Then S is a Banach space and the restriction of any C1
map g on Rk to W , denoted by g|W , is in S. Since x
0 is a snap-back
re-peller of f and all eigenvalues of Df (x0) are greater than one in absolute
value, there exist positive constants λ2, δ1 and a positive integer M such
that λ1 < λ2 < λ0, δ1 < δ0, x−M ∈ B(x0, δ1)\{x0}, det(DfM(x−M)) 6= 0,
x0 ∈ int(fM(B(x0, δ1)\{x0})) and for all g ∈ U (f, λ0− λ2) and x ∈ B(x0, δ1),
constant such that
max{λ2,
λ0+ δ1
1 + δ1
} < λ3 < λ0. (2.2)
Then for any g ∈ UW(f, λ0− λ3), we have that g is one-to-one on B(x0, δ1).
In addition, if x ∈ Rk with |x − x
0|∗ = δ1, by Equation (2.1) with λ1 replaced
by λ3 and Equation (2.2), we get that
|g(x) − x0|∗ > |f (x) − x0|∗− |g(x) − f (x)|∗ > λ3δ1− (λ0− λ3) > δ1.
Moreover, the continuity of g implies that g(B(x0, δ1)) ⊃ B(x0, δ1). Let
V = B(x0, δ1)\{x0} and UW(f, λ0− λ3) = {g|W : g ∈ U (f, λ0− λ3)}. For the
first desired result, we need to show the existence of a snap-back repeller for any g ∈ UW(f, λ0−λ3) near f . Define H : UW(f, λ0−λ3)×W ×V → Rk⊕Rk
by H(g, x, y) = (g(x) − x, gM(y) − x). Then H(f, x0, x−M) = 0 and H is C1
on its domain; refer to [10, Appendix B]. Since all eigenvalues of Df (x0) are
greater than one in absolute value, we have det(Df (x0) − Ik) 6= 0, where Ik
denotes the identity matrix of size k; refer to [29, Lemma V.5.7.2]. By the chain rule, det(DfM(x
−M)) = QMi=1det(Df (x−i)) 6= 0. Hence, by writing
z = (x, y) ∈ W × V , we have det ∂H ∂z (g, z)|g=f, z=(x0,xM) = det Df (x0) − Ik 0 −Ik DfM(x−M) 6= 0; refer to [28, Proposition 0.0]. By the implicit function theorem applied to the function H, there exist positive constants λ4, δ2, η and a C1 map
h : UW(f, λ0 − λ4) → B(x0, δ2) × B(x−M, η) such that λ3 < λ4 < λ0,
δ2 < δ1, B(x−M, η) ⊂ V , B(x0, δ2) ∩ B(x−M, η) = ∅, and for every g ∈
UW(f, λ0− λ4), one has that h(g) ≡ (h1(g), h2(g)) is the unique solution for
the system of equations g(x) = x and gM(y) = x in B(x
0, δ2) × B(x−M, η),
and det(DgM(h2(g))) 6= 0. In particular, h(f ) = (x0, x−M).
To conclude that the point h1(g) is a snap-back repeller of g, it remains
UW(f, λ0 − λ4) and denote y−M +i = gi(h2(g)) for all 0 6 i 6 M − 1. Then
y−M 6= h1(g) and gM(y−M) = h1(g). Since g is one-to-one on B(x0, δ1),
g(B(x0, δ1)) ⊃ B(x0, δ1) and h2(g) ∈ B(x0, δ1), we can define y−M −i =
ˆ
g−i(h2(g)) inductively for i > 1, where ˆg−1 = (g|B(x0, δ1))−1 denotes the
inverse of the restriction of g to B(x0, δ1) and ˆg−i denotes the ith iterate of
ˆ
g−1. Then the sequence {y−i}i∈N forms a backward orbit of h1(g) such that
y−n∈ B(x0, δ1) for all n > M . From Equation (2.1), we obtain that for any
x, y ∈ B(x0, δ1),
|ˆg−1(x) − ˆg−1(y)|∗ < λ−11 |x − y|∗ (2.3)
By considering inequality (2.3) inductively, we have that for any i > 1, |y−M −i− h1(g)|∗ = |ˆg−i(y−M) − ˆg−i(h1(g))|∗ < λ−i1 |y−M − h1(g)|∗.
This shows that limn→∞y−n= h1(g).
Since the norms | · |2 and | · |∗ on Rk are equivalent, the proof of the first
desired result is now complete. The second and third assertions immediately follow from Theorem 2.4 and 2.5.
Notice that from the above proof of Theorem 2.6, it is sufficient to re-quire a smallness of |f − g|2+ ||Df − Dg||2 locally in a neighborhood of the
homoclinic orbit associated to the snap-back repeller, instead of globally in Rk.
As an immediate consequence of the above theorem, we have the following result for a parametrized family.
Corollary 2.7. Let fµ(x) be a one-parameter family of C1maps with variable
x ∈ Rk and parameter µ ∈ R. Assume that f
µ(x) is C1 as a function jointly
of x and µ and that fµ0 has a snap-back repeller. Then for all µ sufficiently
close to µ0, the map fµ has a snap-back repeller, exhibits Li-Yorke chaos,
and has positive topological entropy.
Corollary 2.8. Let f : Rk → Rk be a one-parameter family of C1 maps
with components (f)i(x) = hi(xi) + igi(x) for each 1 6 i 6 k; here we
denote the variable x = (x1, ..., xk) and the parameter = (1, ..., k) in Rk.
If the number of snap-back repellers for each map hi is mi > 1, then for all
sufficiently small ||, the number of snap-back repellers for the map f is at
least QM
i=1mi.
Gardini et al. [6] studied the double logistic map Tλ : R2 → R2 given by
Tλ(x, y) = (1 − λ)x + 4λy(1 − y), (1 − λ)y + 4λx(1 − x)), λ ∈ [0, 1]; (2.4)
therein the basins of attraction of the absorbing areas are determined to-gether with their bifurcations. Moreover, it was mentioned that T12(x, y) = (h2(x), h2(y)), where h(x) = 4x(1 − x), has a snap-back repeller at the origin. Therefore, applying Corollary 2.8, we have the following result.
Corollary 2.9. For all λ near one, the second iterate of system (2.4) has a snap-back repeller, exhibits Li-Yorke chaos, and has positive topological entropy.
3
Covering relations
In this section, we give the background information about covering relations and list some properties of the local Brouwer degree.
3.1
Background and applications
In this subsection, we introduce the definition and some applications of cov-ering relation. Suppose that Rm has a norm | · |. For x ∈ Rm and r > 0, we
denote Bm(x, r) = {z ∈ Rm : |z − x| < r}, that is, the open ball of radius r
centered at the origin 0 in Rm; in short, we write Bm = Bm(0, 1), the open
unit ball in Rm. Moreover, for a subset S of Rm, let S and ∂S denote the
closure and the boundary of S, respectively. It will be always clear from the context which norm is used.
Now, we briefly recall some definitions from [35] concerning covering re-lations.
Definition 3.1. [35, Definition 6] An h-set in Rm is a quadruple consisting of the following data:
• a nonempty compact subset M of Rm,
• a pair of numbers u(M ), s(M ) ∈ {0, 1, ..., m} with u(M ) + s(M ) = m, • a homeomorphism cM : Rm → Rm = Ru(M ) × Rs(M ) with cM(M ) =
Bu(M )× Bs(M ), where S × T is the Cartesian product of sets S and T .
For simplicity, we will denote such an h-set by M and call cM the coordinate
chart of M ; furthermore, we use the following notations: Mc= Bu(M )× Bs(M ), Mc− = ∂B
u(M )× Bs(M ), M+
c = Bu(M )× ∂B s(M ),
A covering relation between two h-sets is defined as follow.
Definition 3.2. [35, Definition 7] Let M, N be h-sets in Rm with u(M ) =
u(N ) = u and s(M ) = s(N ) = s, f : M → Rm be a continuous map, and fc= cN ◦ f ◦ c−1M : Mc → Ru× Rs. We say M f -covers N , and write
M =⇒ N,f if the following conditions are satisfied:
1. there exists a homotopy h : [0, 1] × Mc→ Ru× Rs such that
h(0, x) = fc(x) for x ∈ Mc, (3.1)
h([0, 1], Mc−) ∩ Nc = ∅, (3.2)
h([0, 1], Mc) ∩ Nc+ = ∅; (3.3)
2. there exists a map ϕ : Ru → Ru such that
h(1, p, q) = (ϕ(p), 0) for any p ∈ Bu and q ∈ Bs,
ϕ(∂Bu) ⊂ Ru\Bu; and
3. there exists a nonzero integer w such that the local Brouwer degree deg(ϕ, Bu, 0) of ϕ at 0 in Bu is w; refer to [35, Appendix] for its
properties.
Usually, we will be not interested in the values of w among covering relations and we just write M =⇒ N instead of Mf =⇒ N .f,w
Next, we list two important results derived from the covering relations which is proved by Zgliczy´nski and Gidea in [35]. The first one is that a closed loop of covering relations implies existence of a periodic point.
Theorem 3.3. [35, Theorem 9] Let {fi}ki=1 be a collection of continuous
maps on Rm and {M
and Mi fi
=⇒ Mi+1 for 1 6 i 6 k. Then there exists a point x ∈ int(M1) such
that
fi◦ fi−1◦ · · · ◦ f1(x) ∈ int(Mi+1) for i = 1, ...k, and
fk◦ fk−1◦ · · · ◦ f1(x) = x.
The following one shows that a covering relation is persistent under C0 small perturbations.
Theorem 3.4. [35, Theorem 14] Let M and N be h-sets with u(M ) = u(N ) = u and s(M ) = s(N ) = s and let f, g : M → Rm be continuous. Assume that M =⇒ N and that the coordinate chart cf,w N satisfies a Lipschitz
condition. Then there exists > 0 such that if |f (x) − g(x)| < for all x ∈ M then M =⇒ N .g,w
Moreover, the following one shows that a covering relation is persistent under C0 small perturbations. This result slightly extends theorem 3.4 by
dropping the Lipschitz condition of the coordinate chart.
Proposition 3.5. Let M1 and M2 be h-sets with u(M1) = u(M2) = u and
s(M1) = s(M2) = s and let f , g : M1 → Rm be continuous. Assume that
M1 f,w
=⇒ M2.
Then there exists δ > 0, such that if |f (x) − g(x)| < δ for all x ∈ M1 then
M1 g,w
=⇒ M2.
Proof. By using Theorem 3.4, there exists > 0 such that if |fc(x)−gc(x)| <
for all x ∈ M1,c then
M1 g,w
=⇒ M2.
Since M1 is compact, there exists r > 0 such that f (M1) ⊂ Bm(0, r).
If |f (x) − g(x)| < 1 for all x ∈ M1, then g(M1) ⊂ Bm(0, r + 1). By
uniform continuity of cM2 on Bm(0, r + 1), there exists δ
if z, z0 ∈ Bm(0, r + 1) and |z − z0| < δ0 then |cM2(z) − cM2(z
0)| < . Let
δ = min{δ0, 1}. If |f (x) − g(x)| < δ for all x ∈ M1 then
max x∈M1,c |fc(x) − gc(x)| = max x∈M1 |cM2(f (x)) − cM2(g(x))| < . Thus M1 g,w =⇒ M2.
3.2
Properties of local Brouwer degree
In this subsection, we list some basic properties of local Brouwer degree; refer to [30, Chapter III] for the proof. Let n be a positive integer and T ⊂ Rn be an open and bounded set. Let ϕ : D → Rn be continuous, ¯T ⊂ D and
q ∈ Rn with q /∈ ϕ(∂T ). 1. Integer property:
deg(ϕ, T, q) ∈ Z;
2. Solution property: If deg(ϕ, T, q) 6= 0, then there exists x ∈ T such that
ϕ(x) = q;
3. Invariance under homotopy: Let H : [0, 1] × D → Rn be continuous.
Suppose that p /∈ H([0, 1], ∂T ). Then for all λ ∈ [0, 1], deg(H0, T, p) = deg(Hλ, T, p);
4. Local constant property: If p and q lie in the same connected compo-nent of Rn\ϕ(∂T ), then
deg(ϕ, T, p) = deg(ϕ, T, q); 5. The excision property: Assume ϕ−1(q) ∩ D ⊂ T, then
6. Multiplication property: Let ψ : Rn → Rn be a continuous mappings
and ∆i be the components of Rn\ϕ(∂T ). Then
deg(ψ ◦ ϕ, T, q) =X
∆i
deg(ψ, ∆i, q) deg(ϕ, T, ∆i);
where deg(ϕ, T, ∆i) = deg(ϕ, T, qi) for some qi ∈ ∆i.
7. Addition property: If T = S
i∈ITi, where each Ti is open, ∂Ti ⊂ ∂T ,
and the family {Ti}i∈I are mutually disjoint, then
deg(ϕ, T, q) =X
i∈I
deg(ϕ, D, q);
8. If ϕ is C1 and for each x ∈ ϕ−1(q) ∩ T the Jacobian matrix of ϕ at x,
denoted by Dϕx, is nonsingular, then
deg(ϕ, T, q) = X
x∈ϕ−1(q)∩T
sgn(det Dϕx),
where sgn represents the sign function.
Form the above properties, we can derive the following proposition which is used later.
Proposition 3.6. Let ψ : Rn → Rn be a C1 map and p ∈ Rn such that
ψ−1(p) consists of a single point and lies in a bounded connected component ∆ of Rn\ ϕ(∂T ), and Dψψ−1(p) is nonsingular. Then
deg(ψ ◦ ϕ, T, p) = sgn(det Dψψ−1(p)) deg(ϕ, T, v),
4
Topological dynamics for multidimensional
perturbations
In this section, the topological dynamics for multidimensional perturbations of maps are studied. We investigate the question (#) with the lower dimen-sional map, for cases(i)-(iii), which has positive topological entropy, snap-back repeller, or topologically crossing homoclinicity and for cases (i)-(ii) and (iv), which has covering relations determined by a transition matrix.
4.1
Snap-back repellers and one dimensional maps
In this subsection, we state our result about the topological entropy of mul-tidimensional perturbations of a continuous map f on a lower dimensional phase space, say Rm, to a continuous family of maps F
λ on a high-dimensional
space, say Rm × Rn
, where λ ∈ R` is a parameter, such that at λ = 0, the singular map F0 is one of the cases (ii) and (iii) referred to question (#). The
case (i) with snap-back repeller on the on a lower dimensional phase space is discussed in section 2.
4.1.1 One dimensional maps
First, we state the results for multidimensional perturbations of a one di-mensional maps.
Let f be a continuous map on R. If the singular map F0 depends only
on the phase variable of f (refer to case (ii)), we have the following result. Theorem 4.1. Let Fλ be a one-parameter family of continuous maps on
R × Rn such that Fλ(x, y) is continuous as a function jointly of λ ∈ R` and
(x, y) ∈ R × Rn. Assume that F
0(x, y) = (f (x), g(x)) for all (x, y) ∈ R × Rn,
where f : R → R and g : R → Rn. Then lim inf
For the case when the singular map is locally trapping along the normal direction (refer to case (iii)), we have the following.
Theorem 4.2. Let Fλ be a one-parameter family of continuous maps on
R × Rn such that Fλ(x, y) is continuous as a function jointly of λ ∈ R` and
(x, y) ∈ R×Rn. Assume that F0(x, y) = (f (x), g(x, y)) for all (x, y) ∈ R×Rn,
where f : R → R, g : R × Rn → Rn, and g(R × S) ⊂ int(S) for some
compact set S ⊂ Rn homeomorphic to the closed unit ball in Rn. Then lim infλ→0htop(Fλ) > htop(f ).
In order to prove the above theorems, we need the following lemma, which can be easy derived from [25]; see also Theorem 3.1 of Misiurewicz and Zgliczy´nski in [26]. It says that for continuous interval maps, the positive topological entropy is realized by horseshoes.
Lemma 4.3. Let I be a closed interval in R and f : I → I be a continuous map with a positive topological entropy, i.e. htop(f ) > 0. Then there exist
sequences {sk}∞k=1 and {tk}∞k=1 of positive integers such that for each k ∈ N
there exist sk disjoint closed intervals, N1, ..., Nsk, which are h-sets in R and
satisfy the covering relations Ni
ftk,wi,j
=⇒ Nj with wi,j ∈ {−1, 1} for all 1 6 i,
j 6 k; moreover, one has limk→∞(log(sk)/tk) = htop(f ).
Now we are ready to prove the Theorems 4.1 and 4.2.
Proof of Theorem 4.1. We only need to consider the case when f has a pos-itive topological entropy. Let δ be an arbitrary number such that 0 < δ < htop(f ). From Lemma 4.3, there exist k, p ∈ N such that fk has p
dis-joint closed intervals, denoted by Ni0 = [a2i, a2i+1] for 0 6 i 6 p − 1 with
a0 < · · · < a2p−1, which are h-sets satisfying
Ni0 ftk=⇒ N,wi,j j0 for 0 6 i 6 p − 1 and 0 6 j 6 p − 1, where wi,j = 1 or −1, and log(p)/k > δ.
Set N0 = ∪p−1i=0Ni0. Since g ◦ fk−1 is continuous and N0 is compact, there exists r > 0 such that g ◦ fk−1(N0) ⊂ B
n(0, r). Set Ni = Ni0 × Bn(0, r) for
0 6 i 6 p−1 and N = ∪p−1i=0Ni. Then every Niis an h-set for 0 6 i 6 p−1 and
N is compact in R × Rn. For λ = 0, we have F0k(x, y) = (fk(x), g ◦ fk−1(x)). Hence there are covering relations:
Ni F0k,wi,j
=⇒ Nj for 0 6 i 6 p − 1 and 0 6 j 6 p − 1.
Since Fλk(z) is uniformly continuous on a compact set, say [−1, 1] × N , as a function jointly of λ and z, by using Theorem 3.4 for p2 times while each c
Nj
is linear and satisfies the Lipschitz condition, there exists λ0 > 0 such that
if |λ| < λ0 then we have
Ni Fk
λ,wi,j
=⇒ Nj for 0 6 i 6 p − 1 and 0 6 j 6 p − 1.
Let m be a positive integer and |λ| < λ0. Consider any closed loop
Nα0 Fk λ =⇒ Nα1 Fk λ =⇒ · · · F k λ =⇒ Nαm,
where every αi ∈ {0, 1, ...p − 1} and αm = α0. By using Theorem 3.3, Fλk
has a periodic point x = x(λ) ∈ int(Nα0) such that F
km
λ (x) = x. Since there
are pm choices of such closed loops, Fk
λ has at least pm periodic points in N .
These periodic points provide a (m, )-separated set for Fλk as long as is a positive number less than gaps of Ni0s, i.e. 0 < < min{a2i− a2(i−1)+1 : 1 6
i 6 p − 1}. Since m is arbitrarily chosen, we have htop(Fλk) > log(p) and so
htop(Fλ) > log(p)/k > δ. Therefore, lim infλ→0htop(Fλ) > htop(f ).
The proof of the second main result is the following.
Proof of Theorem 4.2. Define Gλ = (id, c) ◦ Fλ ◦ (id, c)−1, where id denotes
the identity map on R and c is a homeomorphism from S to Bn. Then
the topological entropies of Gλ and Fλ are equal. By applying the above
argument to the family Gλ while the corresponding cM of a covering relation
4.1.2 Higher dimensional maps
In this subsection, we will study the topological entropy for multidimensional perturbations of a higher dimensional map which has a snap-back repeller.
As the result of Theorem 3.3 and 3.4, we shall construct the a closed loop of covering relations for the map. Throughout this subsection, we assume that f : Rm → Rm is a C1 map having a snap-back repeller x
0 associated
with a transverse homoclinic orbit. We shall construct two closed loops of covering relations for f : the first one is from the snap-back repeller to a homoclinic point then back to the repeller, and the second one consists of just one relation Nr
f
=⇒ Nr, where Nr is one of the h-sets in the first closed
loop. Then we use the covering relations approach to prove that f has a positive topological entropy.
Let L be a linearization of f at x0, that is, L(z) = x0 + Df (x0)(z − x0)
for z ∈ Rm. Since all eigenvalues of Df (x0) are greater than one in absolute
value, there exist a norm | · | on Rm and a constant ρ > 1 such that
|Df (x0)z| > ρ|z| for z ∈ Rm. (4.1)
From now on, we keep this norm fixed.
For any r > 0 and x ∈ Rm, we denote the closed ball with the center x and radius r by
N (x, r) = {x} + Bm(0, r).
For any r > 0 we define an h-set Nx,r in Rmas follows: we set Nx,r = N (x, r),
cNx,r(z) = (z − x)/r, u(Nx,r) = m and s(Nx,r) = 0. Since the point x0 is a
fixed point for f and will play a distinguished role in the following, we will write Nr instead of Nx0,r. Next, we define a homotopy from the map f to L,
its linearization at x0, as follows:
It is easy to see that f0(z) = f (z), f1(z) = L(z) and Dfµ(z) = (1−µ)Df (z)+
µDf (x0) for all µ and z. This homotopy will be later used in covering
relations in the vicinity of the snap-back repeller.
First, we show that the size of the repulsion set for snap-back repeller x0
can be chosen uniformly for all fµ for µ ∈ [0, 1].
Lemma 4.4. Let β = (ρ + 1)/2. Then there exists r0 > 0 such that for any
µ ∈ [0, 1], 0 < r 6 r0, z ∈ Nr with |z − x0| = r, the following holds:
|fµ(z) − x0| > βr.
Proof. By using Taylor’s theorem with an integral remainder, we have fµ(z) − x0 = fµ(z) − fµ(x0) = C(z − x0), where C = C(µ, z, x0) = Z 1 0 Dfµ(x0+ t(z − x0))dt.
By Equation (4.2), we get that C − Dfµ(x0) = Z 1 0 (1 − µ)Df (x0+ t(z − x0)) + µDf (x0)dt − Dfµ(x0) = Z 1 0 (1 − µ)[Df (x0+ t(z − x0)) − Df (x0)]dt. (4.3)
Since Df is continuous at x0 and ρ > 1, there exists r0 > 0 such that if
|y − x0| 6 r0 then |Df (y) − Df (x0)| < (ρ − 1)/2. Hence, from Equation
(4.3), we have that for any µ ∈ [0, 1] and z ∈ Bm(x0, r),
|C − Dfµ(x0)| 6 Z 1 0 (1 − µ)|Df (x0+ t(z − x0)) − Df (x0)|dt < Z 1 0 (1 − µ)ρ − 1 2 dt 6 ρ − 1 2 .
Therefore, by using Equation (4.1), we have that for any µ ∈ [0, 1], 0 < r 6 r0, z ∈ Nr with |z − x0| = r,
|fµ(z) − x0| = |C(z − x0)| = |(C − Dfµ(x0) + Dfµ(x0))(z − x0)|
> |Df (x0)(z − x0)| − |(C − Dfµ(x0))(z − x0)|
> ρr −ρ − 1
Throughout the rest of this subsection, we fix the two constants β and r0 as given in Lemma 4.4. In the following, we establish a covering relation
between two h-sets around the snap-back repeller.
Proposition 4.5. Let r and r1 be two numbers satisfying 0 < r 6 r0 and
0 < r1 6 βr. Then the following covering relation holds:
Nr f
=⇒ Nr1.
Proof. Define h(µ, z) = cNr1(fµ(c−1Nr1(z)). We need to check whether all
con-ditions for the covering relation Nr f
=⇒ Nr1. are satisfied. First we deal with
the conditions in the first item of Definition 3.2. Condition (3.1) is implied by f0 = f , Condition (3.2) follows from Lemma 4.4, and since Nr+1 = ∅,
Condition (3.3) is also satisfied.
Next, we define a map A on Rm by A(z) = (r/r
1)Df (x0)z. Then for z ∈ Bm, we have h(1, z) = L(rz + x0) − x0 r1 = Df (x0)(rz) r1 = A(z).
Moreover, from Equation (4.1) it follows that for z ∈ Bm with |z| = 1,
|A(z)| > ρr r1 >
ρr βr > 1.
Since A is linear, from the above equation we have that deg(A, Bm, 0) =
± det(A) 6= 0.
Next, we give a covering relation from the snap-back repeller x0 to points
near x0, which will be homoclinic points near x0 as the result is used later.
Lemma 4.6. Let r > 0, r1 > 0 and z1 ∈ Rm near x0 satisfy that (|z1− x0| +
r1)/β < r < r0. Then
Nr f
Proof. As in the proof of Proposition 4.5, we set h(µ, z) = cNz1,r1(fµ(c−1Nr(z)).
Again, we need to check all conditions for the covering relation Nr f
=⇒ Nz1,r1.
Condition (3.1) is implied by f0 = f , and since Nz+1,r1 = ∅, Condition
(3.3) is also satisfied.
To verify Condition (3.2), observe that it is equivalent to the following one:
fµ(Nr−) ∩ Nz1,r1 = ∅ for µ ∈ [0, 1]. (4.4)
From Lemma 4.4, it follows that for any z ∈ Nr− (hence |z − x0| = r),
|fµ(z) − z1| = |fµ(z) − x0+ x0− z1| > |fµ(z) − x0| − |x0− z1|
> βr − |x0− z1| > |x0 − z1| + r1− |x0− z1| = r1.
This proves Equation (4.4).
It remains to investigate h(1, z). Define a map A on Rm by A(z) =
(rDf (x0)z + x0 − z1)/r1. Then A is affine and for z ∈ Bm,
h(1, z) = L(rz + x0) − z1 r1
= x0 + Df (x0)(rz) − z1 r1
= A(z).
To prove that deg(A, Bm, 0) = det(Df (x0)) = ±1, it is sufficient to show that
the unique solution ˆz = (1/r)Df (x0)−1(z1−x0) of the equation A(z) = 0 is in
Bm. To this end, observe that from Equation (4.1), we have |Df (x0)−1| 6 ρ−1
and hence |ˆz| 6 1 r|Df (x0) −1| · |z 1− x0| 6 |z1− x0| ρr < |z1− x0| + r1 βr < 1.
The following lemma gives a covering relation from a homoclinic point to the snap-back repeller.
Lemma 4.7. Assume that z0 ∈ Rm such that fk(z0) = x0 for some integer
k > 0 and det(Dfk(z
0)) 6= 0. Then there exists R > 0 such that if 0 < r < R
then there is v ≡ v(r) with 0 < v < r0 such that for any 0 < r2 6 v, we have
Nz0,r
fk
Proof. By continuity of f , there is R1 > 0 such that
fk(Bm(z0, R1)) ⊂ Bm(x0, r0).
Define a homotopy as follows: for µ ∈ [0, 1] and z ∈ Bm(z0, R1),
gµ(z) = (1 − µ)fk(z) + µ(Dfk(z0)(z − z0) + x0). (4.6)
Then gµ(z0) = x0 and dgµ(z) = (1 − µ)Dfk(z) + µDfk(z0) for all µ and z.
Since Dfk(z
0) is nonsingular, there is a constant α > 0 such that for any
z ∈ Rm,
|Dfk(z0)z| > α|z|. (4.7)
Next, we show that there exists a positive number R < min{R1, 2r0/α}
such that for all |z − z0| < R and µ ∈ [0, 1], one has
|gµ(z) − x0| >
α
2|z − z0|. (4.8) To this end, we have to modify the proof of Lemma 4.4 a bit. By using Taylor’s theorem with integral remainder, we have
gµ(z) − x0 = gµ(z) − gµ(z0) = C(z − z0), where C = C(µ, z, z0) = Z 1 0 Dgµ(z0+ t(z − z0))dt.
By Equation (4.6), we get that C − Dgµ(z0) = Z 1 0 (1 − µ)Dfk(z0+ t(z − z0)) + µDfk(z0)dt − Dgµ(z0) = Z 1 0 (1 − µ)[Dfk(z0+ t(z − z0)) − Dfk(z0)]dt. (4.9) Since Dfk is continuous at z
0, there exists R > 0 such that if |y − z0| < R
then
|Dfk(y) − Dfk(z
Hence, from (4.9), we have that for any µ ∈ [0, 1] and z ∈ Bm(z0, R), |C − Dgµ(x0)| 6 Z 1 0 (1 − µ)|Dfk(z0+ t(z − z0)) − Dfk(z0)|dt < Z 1 0 (1 − µ)α 2dt 6 α 2.
Therefore, by using Equation (4.7), we obtain that for any µ ∈ [0, 1] and z ∈ Bm(z0, R), |gµ(z) − x0| = |C(z − z0)| = |(C − Dgµ(z0) + Dgµ(z0))(z − z0)| > |Dfk(z0)(z − z0)| − |(C − Dgµ(z0))(z − z0)| > α − α 2 |z − z0| = α 2|z − z0|.
Now we are ready to prove the desired covering relation (4.5). Let r be a number with 0 < r < R and let v = αr/2. Let r2 be a number with
0 < r2 6 v. Since α > 0 and R < 2r0/α, we have 0 < v < r0. We define a
homotopy hµ by
hµ(z) = cNr2(gµ(c−1Nz0,r(z))) for µ ∈ [0, 1] and z ∈ Bm.
The conditions from Definition 3.2 requiring the proof are only Condition (3.2) and deg(h1, Bm, 0) 6= 0 while the others are clear. To verify Condition
(3.2), note that it is equivalent to the following one:
gµ(Nz−0,r) ∩ Nr2 = ∅ for µ ∈ [0, 1]. (4.10)
From Equation (4.8), it follows that for any z ∈ Nz−0,r (hence |z − z0| = r),
one has
|gµ(z) − x0| >
α
2|z − z0| > r2. This proves Equation (4.10). Finally, since
h1(z) =
r r2
we obtain that h1 is a linear isomorphism; therefore
deg(h1, Bm, 0) = det(Dfk(z0)) 6= 0.
The next proposition shows that the existence of a snap-back repeller as defined in Definition 2.2 implies a positive topological entropy. In [2], Blanco Garcia gave the same result based on Marotto’s definition of a snap-back repeller and results in [19]. Here, we give a new proof by using covering relations.
Proposition 4.8. The topological entropy of f is positive.
Proof. Let β and r0 be as given in Lemma 4.4. Since x0is a snap-back repeller
for f , there exists a sequence {x−i}i∈N such that x−1 6= x0, limi→∞x−i = x0
and for all i ∈ N, f (x−i) = x−i+1 and det(Df (x−i)) 6= 0. Thus, there is
an integer k > 0 such that x−k ∈ B(x0, r0). By the chain rule, we have
det(Dfk(x
−k)) 6= 0. Furthermore, from Lemma 4.7, there exist positive
constants rk and rb such that rb < r0 and
B(x−k, rk) ⊂ B(x0, r0), (4.11)
Nx−k,rk ∩ Nrb = ∅, (4.12)
Nx−k,rk
fk
=⇒ Nrb. (4.13)
Since β > 1, there exists the minimal positive integer a such that βarb >
|x−k− x0| + rk. By the minimum of a and Equation (4.11), we have βa−1rb 6
|x−k − x0| + rk < r0. From Proposition 4.5 and Lemma 4.6, it follows that
we have the following chain of covering relations: Nrb f =⇒ Nβrb f =⇒ · · ·=⇒ Nf βa−1r b f =⇒ Nx−k,rk. (4.14)
Moreover, from Proposition 4.5, it also follows that Nrb
f
These covering relations are enough to produce symbolic dynamics and a positive topological entropy as follows. Let w = max(a, k). It is sufficient to construct an f2w-invariant set on which f2w can be semi-conjugated onto
the shift map σ : Σ+2 → Σ+2, where Σ+2 = {0, 1}N, the one-sided shift space
on two symbols with the standard Tikhonov (product) topology. By using Equations (4.13)-(4.15), one can consider the following chains of covering relations, each one of length 2w (which is counted by the number of iterates of f ): Nrb f =⇒ Nrb f =⇒ Nrb f =⇒ · · ·=⇒ Nf rb, Nrb f =⇒ Nrb f =⇒ · · ·=⇒ Nf rb f =⇒ Nβrb f =⇒ · · ·=⇒ Nf βa−1r b f =⇒ Nx−k,rk, Nx−k,rk fk =⇒ Nrb f =⇒ Nrb f =⇒ · · ·=⇒ Nf rb, Nx−k,rk fk =⇒ Nrb f =⇒ · · ·=⇒ Nf rb f =⇒ Nβrb f =⇒ · · ·=⇒ Nf βa−1r b f =⇒ Nx−k,rk.
Let us denote N0 = Nrb and N1 = Nx−k,rk. Then N0 and N1 are disjoint due
to Equation (4.12). Define Z to be the set of points whose forward orbits under f2w stay in N
0∪ N1, that is,
Z = {z ∈ N0∪ N1 : f2iw(z) ∈ N0∪ N1 for all i ∈ N}.
Then Z is compact. On Z we define a projection π : Z → Σ+2 by π(z)i = j if and only if f2iw(z) ∈ Nj.
It is obvious that the map π is continuous and we have a semiconjugacy: π ◦ f2w = σ ◦ π.
Finally, we shall show that π is onto. This gives us that the topological entropy of f2w on Z is greater than or equal to log 2. Let α = (α0, ..., αl−1) ∈
{0, 1}l for some positive integer l. By a suitable concatenation of the above
listed chains of covering relations and from Theorem 3.3, it follows that there exists a point xα ∈ Nα0 such that
f2iw(xα) ∈ Nαi for 0 6 i 6 l − 1,
It is clear that xα ∈ Z and π(xα) = (α, α, ...) ∈ Σ+2. Since α is arbitrarily
chosen, the set π(Z) contains all repeating sequences. From the density of repeating sequences in Σ+2, it follows that π(Z) = Σ+2.
Now, we list our main results about the multidimensional perturbations of a higher dimensional map which has a snap-back repeller. First, if the singular map depends only on the phase variable of a snap-back repeller, we have the following result.
Theorem 4.9. Let Fλ be a one-parameter family of continuous maps on
Rm × Rn such that Fλ(x, y) is continuous as a function jointly of λ ∈ R`
and (x, y) ∈ Rm× Rn. Assume that F
0(x, y) = (f (x), g(x)) for all (x, y) ∈
Rm × Rn, where f : Rm → Rm is C1 and has a snap-back repeller and g : Rm → Rn. Then F
λ has a positive topological entropy for all λ sufficiently
close to 0.
When the singular map is locally trapping along the normal direction, we have the following.
Theorem 4.10. Let Fλ be a one-parameter family of continuous maps on
Rm × Rn such that Fλ(z) is continuous as a function jointly of λ ∈ R` and
(x, y) ∈ Rm × Rn. Assume that F
0(x, y) = (f (x), g(x, y)) for all (x, y) ∈
Rm × Rn, where f : Rm → Rm is C1 and has a snap-back repeller, g : Rm × Rn → Rn, and g(Rm × S) ⊂ int(S) for some compact set S ⊂ Rn homeomorphic to the closed unit ball in Rn. Then Fλ has a positive topological
entropy for all λ sufficiently close to 0.
Now, we begin to prove Theorem 4.9 and 4.10.
Proof of Theorem 4.9. From the proof of Proposition 4.8, we have a positive integer a such that the following closed loop of covering relations holds:
Nrb f =⇒ Nrb f =⇒ Nβrb f =⇒ · · ·=⇒ Nf βa−1r b f =⇒ Nx−k,rk fk =⇒ Nrb,
By adding the normal direction to the above h-sets and using the persis-tence of covering relation, we shall construct a closed loop of covering re-lations for Fλ, similar to the above loop for f . Recall that the singular
map F0 is of the form F0(x, y) = (f (x), g(x)) ∈ Rm × Rn. Set N =
(∪a−1i=0Nβir
b) ∪ (∪
k
i=0fi(Nx−k,rk)). Since g is continuous and N is compact,
there exists r > 0 such that g(N ) ⊂ Bn(0, r). Let us define the
correspond-ing h-sets in Rm × Rn as follows. For i = 0, 1, ..., a − 1, we define h-sets
Nβ0ir b in R m × Rn by N0 βir b = Nβirb × Bn(0, r), u(N 0 βir b) = m, s(N 0 βir b) = n and cN0
βirb(x, y) = (cNβirb(x), y/r). Moreover, we define an h-set N
0 x−k,rk in Rm× Rnby Nx0−k,rk = Nx−k,rk× Bn(0, r), u(N 0 x−k,rk) = m, s(N 0 x−k,rk) = n and cN0 x−k,rk(x, y) = (cNx−k,rk(x), y/r).
Observe that we have the following closed loop of covering relations for F0.
Lemma 4.11. The following covering relations hold: Nr0 b F0 =⇒ Nr0 b F0 =⇒ Nβr0 b F0 =⇒ · · ·=⇒ NF0 β0a−1r b F0 =⇒ Nx0−k,r k F0k =⇒ Nr0 b,
Proof of Lemma 4.11. For each covering relation under consideration N0 F
j 0 =⇒ M0 with j = 1 or k, a homotopy ˆh : [0, 1] × Bm× Bn→ Rm+n by ˆ h(µ, x, y) = h(µ, x),1 − µ r g ◦ f j−1(c−1 N (x)) ,
where h is the homotopy from the corresponding covering relation N f
j =⇒ M . Then we have ˆ h(0, x, y) = (h(0, x)) ,1 rg ◦ f j−1(c−1 N (x)) = cM ◦ fj◦ c−1N (x), 1 rg ◦ f j−1(c−1 N (x)) = (F0j)c(x, y).
Since ˆh([0, 1], N0,−) ⊂ h([0, 1], N−) × Rn, we get that Condition (3.2) in
Definition 3.2 follows from the analogous Condition for h. Condition (3.3) is satisfied due to
ˆ
Finally, note that
ˆ
h(1, x, y) = (h(1, x), 0).
Therefore, the other conditions in Definition 3.2 are also satisfied.
From Theorem 3.4, there exists λ0 > 0 such that if |λ| < λ0 then the
following chain of covering relations holds for Fλ:
Nr0 b Fλ =⇒ Nr0 b Fλ =⇒ Nβr0 b Fλ =⇒ · · · Fλ =⇒ Nβ0a−1r b Fλ =⇒ Nx0 −k,rk Fk λ =⇒ Nr0 b, (4.16)
Similar to the proof of Proposition 4.8, covering relations listed in (4.16) are sufficient to produce the symbolic dynamics and a positive topological entropy for Fλ with |λ| < λ0. This completes the proof of Theorem 4.9.
Proof of Theorem 4.10. Define Gλ = (id, c) ◦ Fλ◦ (id, c)−1, where id denotes
the identity map on Rk and c is a homeomorphism from S to Bn. Then the
conclusion follows from the above argument applied to Gλ.
4.2
Topologically crossing homoclinicity
In this subsection, we discuss the topological entropy for multidimensional perturbations of topologically crossing homoclinicity.
4.2.1 Background
First, we introduce some definition and results. Let f : Rm → Rm be a
diffeomorphism with a hyperbolic periodic point p at which the stable and unstable subspaces have dimensions u and s, respectively. Let | · | be a norm on Rm. The stable and unstable manifolds of p are defined to be Ws(p) =
{x ∈ Rm : |fn(x) − fn(p)| → 0 as n → ∞} and Wu
(p) = {x ∈ Rm : |fn(x) − fn(p)| → 0 as n → −∞}, respectively. The deleted stable and unstable
manifold of p are given by ˆWs(p) = Ws(p)\{p} and ˆWu(p) = Wu(p)\{p},
of p. For nonempty subsets A, B of Rm, we denote d(A, B) = inf{|x − y| : x ∈ A and y ∈ B}. Here, we are mainly concern the case when ˆWs(p) and
ˆ
Wu(p) has a topologically crossing intersection which is defined as follows.
Definition 4.12. [8, Definition 3] Consider Rm as an m-dimensional ori-ented manifold, and let Wu and Ws be two oriented C1 submanifolds of Rm
with dimensions u and s, respectively, such that u + s = m. We say that Wu and Ws have a topologically crossing intersection if there are compact embedded C1 submanifold Vu of Wu and Vs of Ws with dimensions u and s
and with boundaries ∂Vu and ∂Vs(with respect to Wu and Ws), respectively,
such that
1. ∂Vu∩ Vs= Vu∩ ∂Vs= ∅;
2. For every 0 < ε < min{d(∂Vu, Vs), d(Vu, ∂Vs)}, there exists a homo-topy h : [0, 1] × Rm → Rm satisfying the following:
(a) h(0, x) = x for all x ∈ Rm and the map x 7→ h(1, x) is an embed-ding;
(b) |h(t, x) − x| < ε for all x ∈ Vu∪ Vs and all t ∈ [0, 1];
(c) h(1, Vu) and Vs are transverse submanifolds; and
(d) the oriented intersection number of h(1, Vu) and Vs, denoted by I(h(1, Vu), Vs), is nonzero, where I(A, B) for two oriented
sub-manifolds A and B of Rm with dim A + dim B = m is defined
by
I(A, B) = X
x∈A∩B
Ix(A, B),
and Ix(A, B) is +1 or −1 depending on whether the orientation
induced on TxA⊕TxB agrees or not with the orientation on TxRm, respectively.
In this case, the submanifolds Vu and Vs will be referred as a good pair for the topological crossing between Wu and Ws.
There is a relation between a topological crossing and the local Brouwer degree. Let Vu and Vs be a good pair for a topological crossing intersection between oriented submanifolds Wu and Ws of an oriented manifold W with
dim(W ) = m, dim(Wu) = u, dim(Ws) = s, and u + s = m. Assume that
there exist a closed neighborhood U of Vu∩ Vs in W and local coordinates
(x, y) on U such that Vu ⊂ U, Vs ⊂ U, U = B
u × Bs, Vs = {(x, y) ∈
Bu× Bs : x = 0}, and Vu = {ψ(x) ∈ Bu × Bs : x ∈ Bu}, where ψ is a C1
parametrization of Vu. Let πu : Ru × Rs → Ru be the projection given by
πu(x, y) = x. The following lemma says the local Brouwer degree and the
oriented intersection number are identical.
Lemma 4.13. [8, Lemma 3] Under the above assumptions and notations, we have that (i) in Ru, the origin 0 /∈ π
u(ψ(∂Bu)); (ii) deg(πu◦ ψ, Bu, 0) is well
defined; and (iii) deg(πu ◦ ψ, Bu, 0) = I(h(1, Vu), Vs), where I(h(1, Vu), Vs)
is the oriented intersection number of h(1, Vu) and Vs for any homotopy h as given in Definition 4.12.
4.2.2 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 Fλ be a one-parameter family of continuous maps on
Rm such that Fλ(x) is continuous as a function jointly of λ ∈ R` and x ∈ Rm,
where λ is a parameter. Assume that F0(x) = f (x) for all x ∈ Rm, where
f : Rm → Rm is a C1 diffeomorphism with a hyperbolic periodic point which
N > 0 and a number λ0 > 0 such that both f and Fλ 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 Fλ be a one-parameter family of continuous maps on
Rm × Rk such that Fλ(x, y) is continuous as a function jointly of λ ∈ R`,
x ∈ Rm and y ∈ Rk, where λ is a parameter. Assume that F0(x, y) =
(f (x), g(x)) ∈ Rm× Rk for all x ∈ Rm and y ∈ Rk, where f : Rm → Rm is a
C1 diffeomorphism with a hyperbolic periodic point which has a topologically
crossing homoclinic point, and g : Rm → Rk is a continuous function. Then
there exist an integer N > 0 and a number λ0 > 0 such that both f and Fλ
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
Rm × Rk such that Fλ(x, y) is continuous as a function jointly of λ ∈ R`,
x ∈ Rm and y ∈ Rk, where λ is a parameter. Assume that F
0(x, y) =
(f (x), g(x, y)) ∈ Rm× Rk
for all x ∈ Rm and y ∈ Rk, where f : Rm → Rm is
a C1 diffeomorphism with a hyperbolic periodic point which has a topologically crossing homoclinic point, and g : Rm× Rk → Rk is continuous on Rm× S
and g(Rm × S) ⊂ int(S) for some compact set S ⊂ Rk homeomorphic to
the closed unit ball in Rk. Then there exist an integer N > 0 and a number
λ0 > 0 such that both f and Fλ 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 Wu(p) and s = dim Ws(p). Since ˆWu(p) and ˆWs(p) have a topologically crossing
matrix Dfp of f at p preserves the splitting Rm = Ru⊕ Rs. By the
Hartman-Grobman Theorem, there exist a closed neighborhood U of p and a homeo-morphism ϕ of U into Rm 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 Rm. Thus f is a linear map on U . Write
f (x, y) = (Lux, Lsy) for (x, y) ∈ U, where Lu is a u × u matrix with all
eigenvalues greater than one in absolute value and Ls is an s × s matrix with
all eigenvalues less than one in absolute value. There exist norms | · |u and
| · |s on Ru and Rs, respectively, and constants ρ1 > 1 and 0 < ρ2 < 1 such
that
|Lux|u > ρ1|x|u and |Lsy|s 6 ρ2|y|s for x ∈ Ru and y ∈ Rs. (4.17)
Since all norms on Rm are equivalent, we may assume U = B
u × Bs and
define the norm | · | on Rm to be the maximum norm of the norms | · |u and
| · |s on Ru and Rs. 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 Wu(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 ⊂ Rm is defined to be
sup{|x − y| : x, y ∈ E}.
Proof. Since V ⊂ Wu(p), there exists a positive integer n
1such that f−n1(V ) ⊂
that K > sup{kDfn1
z k : 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 Ru× Rs on U , Equations (4.17)
and (4.18) imply f−n1−n2(V ) ⊂ f−n2(B
u × {0}) ⊂ Bu(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) = (Ln2
u (x), Lns2(y)) ∈ {Lnu2(x)} × Bs(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)| = |Ln2
s (y1) − Lns2(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 ˆWu(p) and ˆWs(p) have a topological crossing
in-tersection, there exist a point q 6= p and two compact embedded submanifolds Vu of ˆWu(p) and Vs of ˆWs(p) such that Vu and Vs form a good pair, and
q ∈ Vu∩ Vs. We may assume that both sets Vu and Vs are in int(U ), and
Vu has no intersection with the subspace Ru× {0}, based on the following lemma.
Lemma 4.18. For any sufficiently integer n ∈ N, there exist submanifolds Vu
n of ˆWu(p) and Vnu of ˆWs(p) such that Vnu and Vns form a good pair with the
same oriented intersection number as good pair Vu and Vs, Vnu ⊂ fn(Vu),
Vs
n ⊂ fn(Vs), and Vnu∪ Vns ⊂ int(U ).
Proof. First, we show that fn(Vu) and fn(Vs) form a good pair for n ∈ N.
Since f is a C1 diffeomorphism, fn(Vu) and fn(Vs) are compact embedded
C1 submanifolds of ˆWu(p) and ˆWs(p), respectively. Since Vu ⊂ ˆWu(p), Vs ⊂ ˆWs(p), and ∂Vu∩Vs= Vu∩∂Vs = ∅, we also have ∂fn(Vu)∩fn(Vs) =
