Topological transitivity and sensitive dependence on initial conditions in discrete

Topological transitivity and sensitive dependence on initial conditions in discrete

dynamical systems

JUNKOKASE ANDSHINTARONAKAO Department of Mathematics, Kanazawa University

Kanazawa 920-1192, Japan

(Received January 5, 2010 and accepted in revised form February 12, 2010)

Abstract Let(X,d)be a complete separable metric space without isolated points.

Let{fⁿ}_n∈NNN^∗ and{fⁿ}n∈ZZZbe the dynamical systems defined by a continuous map f:X→Xand a homeomorphism f:X→Xrespectively, whereNNN^∗=NNN∪ {0}. We show that if the dynamical system{fⁿ}n∈ZZZis topologically transitive, both{fⁿ}n∈NNN^∗

and{f⁻ⁿ}_n∈NNN^∗ are topologically transitive. Moreover, we show that if the dynami- cal system{fⁿ}n∈ZZZis topologically transitive and has sensitive dependence on initial conditions, at least one of the dynamical system{fⁿ}_n∈NNN^∗ and{f⁻ⁿ}_n∈NNN^∗ have sen- sitive dependence on initial conditions.

Key words and Phrases. Topological transitivity, sensitive dependence on initial conditions, dynamical systems, f-subinvariant functions,f-invariant functions 2000Mathematics Subject Classification. 37B99, 54H20

1 Introduction

Let(X,d)be a metric space andfa continuous map fromXtoX. Let us consider the dynamical system{fⁿ}_n∈NNN^∗defined by f. Here fⁿdenotes thentimes composite map of f, and the parameternruns over the set of all nonnegative integersNNN^∗=NNN∪ {0}. The dynamical system {fⁿ}n∈NNN^∗ is said to be topologically transitive if for any nonempty open setsUandV there is an integerk≥1 such that f^k(U)∩V is nonempty. Also, the dynamical system{fⁿ}n∈NNN^∗ is said to have sensitive dependence on initial conditions if there exists a constantδ >0 such that, for anyx∈X and for any neighborhoodU ofx withU≥2, there exist a pointy∈Uand an integern≥0 satisfyingd(fⁿ(x),fⁿ(y))>

δ, whereU is the number of elements of the setU.

These two properties play an important role in the theory of chaos. In fact, Devaney [4]defined the dynamical system{fⁿ}n∈NNN^∗ to bechaoticif it is topologically transitive

and has sensitive dependence on initial conditions, and the periodic points of f are dense inX. The pioneering work by Li and Yorke[5]asserts that the dynamical system defined by a continuous map f on an interval is chaotic in this sense if f has three periodic points. Furthermore, J. Banks et al.[2]showed that if a continuous map f is topologically transitive and has dense periodic points, then f has sensitive dependence on initial conditions.

If the given map f is a homeomorphism, these two properties are defined also for dynamical system{fⁿ}n∈ZZZ, where the parameter nruns over the set of all integers ZZZ.

However, the relationship between these properties for {fⁿ}_n∈NNN^∗ and{fⁿ}n∈ZZZ has not been well understood. The aim of this paper is to clarify this problem. We show that if the dynamical system{fⁿ}n∈ZZZdefined by a homeomorphism f is topologically tran- sitive, then both {fⁿ}_n∈NNN^∗ and{f⁻ⁿ}_n∈NNN^∗ are topologically transitive (Theorem 4.1).

Moreover, we show that if the dynamical system{fⁿ}n∈ZZZis topologically transitive and has sensitive dependence on initial conditions, then at least one of the dynamical sys- tems{fⁿ}_n∈NNN^∗ and{f⁻ⁿ}_n∈NNN^∗has sensitive dependence on initial conditions (Theorem 4.2). The idea of their proofs is to give new characterizations of topological transitivity and sensitive dependence on initial conditions, which constitutes the main part of the paper.

The paper is organized as follows. In section 2, we introduce new characterizations for topological transitivity of{fⁿ}_n∈NNN^∗ and{fⁿ}n∈ZZZ. They are based on the use of the class of upper semicontinuous and f-subinvariant (resp. f-invariant) functions, which we denote byΓsandΓrespectively (Definitions 2.2 and 2.4). We prove that the dynam- ical system{fⁿ}_n∈NNN^∗is topologically transitive if and only if anyγ∈Γshas a minimum and the set M_γ :={x∈X | γ(x) =minγ} is dense inX, where minγ =min_y∈Xγ(y) (Theorem 2.2). We can also prove a characterization of the topological transitivity of the dynamical system{fⁿ}n∈ZZZby using the class of functionsΓ(Theorem 2.4).

In section 3, we discuss the sensitive dependence on initial conditions for{fⁿ}_n∈NNN^∗

and{f⁻ⁿ}n∈NNN^∗. We construct some functionsr₊∈Γsandr∈Γ, and use them to prove thatδ >0 in the definition of sensitive dependence on initial conditions can be taken to be dependent onx∈Xif both{fⁿ}n∈NNN^∗ and{fⁿ}n∈ZZZhave sensitive dependence on ini- tial conditions (Theorem 3.1, Theorem 3.2). In section 4, we prove the main theorems (Theorem 4.1 and Theorem 4.2). We hope to construct an example of a homeomor- phism f on a complete separable metric space(X,d)without isolated points such that dynamical systems{fⁿ}n∈NNN^∗ and{f⁻ⁿ}_n∈ZZZ are topologically transitive, but that either {fⁿ}_n∈NNN^∗ or{f⁻ⁿ}_n∈NNN^∗ has sensitive dependence on initial conditions. It is a future problem to construct such examples.

2 Topological transitivity in dynamical systems

(1) Topological transitivity in discrete dynamical systems with the param- eterNNN^∗=NNN∪ {0}

Let(X,d)be a metric space and f a continuous map fromX toX. In this section we consider the dynamical system {fⁿ}_n∈NNN^∗ defined by a continuous map f in(X,d), where the parameter runs over the set of all nonnegative integers N^∗=NNN∪ {0}. For x∈X, the set{x,f(x),f²(x),···}is called apositive orbit of fand denoted byO₊(f;x), furthermore the set {f(x),f²(x),···}is denoted by O₊(f;x). Also, we denote byD₊ the set of pointsx∈X for whichO₊(f;x)is dense inX.

Definition 2.1. Let(X,d)be a metric space and f a continuous map fromX toX. The dynamical system {fⁿ}n∈NNN^∗ is said to betopologically transitive if for any nonempty open setsU andV there is an integerk≥1 such that f^k(U)∩V is nonempty.

Theorem 2.1 ([1]). Let(X,d)be a complete separable metric space and f a continuous map from X to X. The following three conditions are mutually equivalent.

(1) The dynamical system{fⁿ}_n∈NNN^∗ is topologically transitive.

(2) There is a point x∈X such that the orbit O₊(f;x)is dense in X.

(3) The set{x∈X |O₊(f;x) =X}is dense in X.

In particular if X has no isolated points, then the above three conditions are also mutu- ally equivalent to the following (4) or (5).

(4) There is a point x∈X such that the positive orbit O₊(f;x)is dense in X.

(5) The set{x∈X |O₊(f;x) =X}is dense in X.

If we assume only thatX is a topological space, Theorem 2.1([1]) is proved by using Baireʼs category theorem ([6]), axiom of separation, second axiom of countability. Let X be a topological space andX_n(n∈NNN)an arbitrary sequence of closed sets ofX. The spaceXis calledBaire spaceorXsatisfies Baireʼs category theorem if at least one of the X_nhas an inner point provided that^0∞_n=1X_n has an inner point. If(X,d)is a complete metric space, X is a Baire space and Baireʼs category theorem holds inX. Moreover if X is a Baire space andY is an open subset ofX, thenY is also a Baire space as a subspace ([3]). Theorem 2.1 is well known and characterizes the topological transitivity by the existence of an x∈X such that the orbitO₊(f;x) is dense in X. Theorem 2.2 below gives a new characterization of topological transitivity in dynamical systems.

Definition 2.2. Let (X,d) be a metric space, γ a function from X to[0,∞), and f a continuous map fromX toX. γis said to be f -subinvariant(resp. f -superinvariant) if

γ(x)≤γ(f(x))(resp. γ(x)≥γ(f(x))) holds for anyx∈X. The class of functionsγis defined as follows:

Γs=

def{γ:X→[0,∞)|γis f-subinvariant, upper semicontinuous}

By using this class of functions, we give a necessary and sufficient condition for the dynamical system{fⁿ}n∈NNN^∗ to be topologically transitive.

Theorem 2.2. Let(X,d)be a complete separable metric space without isolated points and f a continuous map from X to X. Then the following two conditions are mutually equivalent.

(1) The dynamical system{fⁿ}_n∈NNN^∗ is topologically transitive.

(2) For any γ∈Γs, it has a minimum and the set M_γ :={x∈X |γ(x) =minγ}is dense in X, whereminγ=miny∈Xγ(y).

Moreover when the condition (1) or (2) holds, wefind D₊=¹_γ∈ΓsM_γ.

To show Theorem 2.2, we prepare three Lemmas: Lemma 2.1,Lemma 2.2,Lemma 2.3.

Lemma 2.1. Let(X,d)be a complete metric space without isolated points. Assume that the dynamical system{fⁿ}_n∈NNN^∗ is topologically transitive. Then for anyγ∈Γs, it has a minimum and the set M_γ:={x∈X |γ(x) =minγ}is dense in X. Furthermore the inclusion relation D₊⊂M_γ holds for anyγ∈Γs.

Proof. Setm=infx∈Xγ(x). Obviously,m<∞andm≥0. Set Xn=

x∈X

γ(x)≥m+1 n

(n≥1).

X_nis a closed set. We assume thatγdoes not have a minimum. It is clearX=^0∞_n=1X_n. From Baireʼs category theorem, there exists a nonempty open setOsuch thatO⊂Xn0

for somen₀∈NNN. Since there is anx₀∈X such that the orbitO₊(f;x₀) is dense inX from Theorem 2.1, there isl≥0 satisfying f^l(x₀)∈O. Also, we have

m+ 1

n₀ ≤γ(f^l(x0))≤γ(f^k(x0)) for k≥l.

BecauseXhas no isolated points,{f^l(x₀),f^l+1(x₀),···}is dense subset ofX. From this and the upper semicontinuity ofγ,

γ(x)≥m+ 1

n₀ (x∈X).

So we havem≥m+_n¹₀. This is a contradiction and thereforeγ has a minimum. To show that M_γ is dense in X, it is sufficient to prove the inclusion relation D₊ ⊂M_γ. We assume that there exists x₁∈D₊ such that x₁∈/M_γ. It is clear that m<γ(x₁)≤ γ(f(x1))≤γ(f²(x1))≤ · · ·. The denseness ofO₊(f;x1)and the upper semicontinuity ofγ yieldγ(x)≥γ(x₁)>mfor anyx∈X. This is a contradiction to the fact thatmis

minimum ofγ. Therefore we have thatD₊⊂M_γ.

Lemma 2.2. Let(X,d)be a complete separable metric space, f a continuous map from X to X, and{U_n}a countable basis of X which is nonempty for n≥1. We define a set F_m⁽ⁿ⁾and a mapκ : X→[0,∞)as follows:

F_m⁽ⁿ⁾ := f^−m(U_n^c) (m≥0,n≥1), κ(x) :=

∑

n=1

1 3ⁿ·

∏

m=0

111Fm⁽ⁿ⁾(x) (x∈X).

Thenκ∈Γsand M_κ=D₊.

Proof. SinceFm⁽ⁿ⁾(m≥0, n≥1)is a closed set ofX, its indicator function 111

Fm⁽ⁿ⁾(x): X →[0,1]is upper semicontinuous. Also

∏

111Fm⁽ⁿ⁾(x) = lim

k→∞

∏

111Fm⁽ⁿ⁾(x) is upper semicontinuous function. Set a functiong_n(x)as follows:

gn(x):=

∏

m=0

111_F(n) m (x).

We will showκ(x)is upper semicontinuous. For anyx0∈X,ε>0, there is anN∈NNN such that∑^∞n=N+1 1

3ⁿ <ε. Hence we have κ(x)<

∑

n=1

1

3ⁿ·g_n(x) +ε for any x∈X.

Because∑^N_n=13¹ⁿ·g_n(x) is upper semicontinuous atx₀, there exists an open neighbor- hoodUofx₀satisfying the following inequality.

∑

1

3ⁿ·g_n(x)<

∑

n=1

1

3ⁿ·g_n(x₀) +ε for x∈U. From this,

κ(x) <

∑

n=1

1

3ⁿ·g_n(x) +ε

≤ κ(x₀) +2ε (x∈U).

So we see thatκis upper semicontinuous. Obviously 111_F(n)

m (f(x)) =111_F(n)

m+1(x), thus κ(f(x)) =

∑

n=1

1 3ⁿ·

∏

m=0

111_F(n) m (f(x))

≥

∑

n=1

1 3ⁿ·

∏

m=0

111Fm⁽ⁿ⁾(x) =κ(x).

Therefore we haveκ∈Γs. Moreover, the following equivalent relations hold.

x∈M_κ ⇐⇒ κ(x) =0

⇐⇒

∑

n=1

1 3ⁿ·

∏

m=0

111Fm⁽ⁿ⁾(x) =0

⇐⇒

∏

m=0

111Fm⁽ⁿ⁾(x) =0 for n≥1

⇐⇒ ^∀n≥1, ^∃m=m(n)≥0, s.t. f^m(x)∈U_n

⇐⇒ O₊(f;x) =X

⇐⇒ x∈D₊

Consequently,M_κ=D₊.

Lemma 2.3. Let(X,d)be a complete separable metric space without isolated points and f a continuous map from X to X. Assume that for anyγ ∈Γs, it has a minimum and the set M_γ :={x∈X |γ(x) =minγ} is dense in X. Then the dynamical system {fⁿ}_n∈NNN^∗ is topologically transitive.

Proof. Letκ :X →[0,∞)be a function defined on Lemma 2.2. Since we knowκ∈Γs

from Lemma 2.2,κhas a minimum and the setM_κ:={x∈X|κ(x) =minκ}is dense in X. Put minκ=a. Clearlya≥0. We assumea>0. There is a uniqueα1,α2,··· ∈ {0,1} such thata=∑^∞n=1αn

3ⁿ. Putl=min{n∈NNN |αn=1}. M_κ is contained byU_l^c, which impliesU_l∩M_κ= /0. This contradicts to the denseness ofM_κ inX. Therefore we have a=0. From Lemma 2.2,M_κ=D₊. SinceM_κis dense inX,D₊is dense inX. It follows from Theorem 2.1 that the dynamical system{fⁿ}n∈NNN^∗ is topologically transitive.

Proof of Theorem 2.2. By Lemma 2.1, (1) implies (2) and the inclusion relationD₊⊂ 1γ∈ΓsM_γ holds. Lemma 2.3 shows that (2) implies (1). Furthermore, sinceκ∈Γsand M_κ=D₊, we have the inclusion¹_γ∈ΓsM_γ⊂D₊.

(2) Topological transitivity in discrete dynamical system with the parame- terZZZ

Let(X,d)be a metric space andf a homeomorphism fromX toX. We consider the case when the parameter runs over the set of all integersZZZ. The set{···f⁻²(x),f⁻¹(x),x}

is called anegative orbit of f, and is denoted byO₋(f;x).

Also{···,f⁻²(x),f⁻¹(x),x,f(x),f²(x)···}is called anorbit of f and is denoted by O(f;x). ObviouslyO₊(f⁻¹;x) =O₋(f;x). We denote byDandD₋ the sets of points x∈X for whichO(f;x)andO₋(f;x)are dense inX respectively.

Definition 2.3. Let(X,d)be a metric space and f a homeomorphism fromX toX. The dynamical system{fⁿ}n∈ZZZ istopologically transitiveif for any nonempty open setsU andV there exists an integern∈ZZZsuch that fⁿ(U)∩V is nonempty.

The following Theorem 2.3 is well known.

Theorem 2.3([1]). Let(X,d)be a complete separable metric space and f a homeomor- phism from X to X. Then the following three conditions are mutually equivalent.

(1) The dynamical system{fⁿ}_n∈ZZZis topologically transitive.

(2) There is a point x∈X such that the orbit O(f;x)is dense in X.

(3) The set{x∈X |O(f;x) =X}is dense in X.

Definition 2.4. Let γ be a function from X to [0,∞). γ is said to be f -invariant if γ(x) =γ(f(x))holds for anyx∈X. We define the classΓof functionsγ as follows:

Γ=

def{γ:X→[0,∞)|γis f-invariant, upper semicontinuous}

Theorem 2.4. Let(X,d)be a complete separable metric space and f a homeomorphism from X to X. Then the following two conditions are mutually equivalent.

(1) The dynamical system{fⁿ}n∈ZZZis topologically transitive.

(2) For anyγ∈Γ, it has a minimum and the set M_γ:={x∈X|γ(x) =minγ}is dense in X.

Moreover if the condition (1) or (2) holds, then D=¹_γ∈ΓM_γ holds.

To show Theorem 2.4, we prepare three lemmas: Lemma 2.4, Lemma 2.5, Lemma 2.6.

Lemma 2.4. Let(X,d)be a complete separable metric space and f a homeomorphism from X to X. Suppose that the dynamical system {fⁿ}n∈ZZZ is topologically transitive.

Then for anyγ∈Γ, it has a minimum and the set M_γ:={x∈X |γ(x) =minγ}is dense in X. Also, the inclusion D⊂M_γ holds.

Lemma 2.5. Let(X,d) be a complete separable metric space, f a homeomorphism from X to X and {U_n} a countable basis of X which is nonempty for any n≥1. We define a set F_m⁽ⁿ⁾and a mapκ: X→[0,∞)as follows:

F_m⁽ⁿ⁾ := f^−m(U_n^c) (m∈ZZZ,n∈NNN), κ(x) :=

∑

n=1

1

3ⁿ·_m

∏

Thenκ∈Γand M_κ=D.

Lemma 2.6. Let(X,d)be a complete separable metric space and f a homeomorphism from X to X. Suppose that for any γ ∈Γ, it has a minimum and the set M_γ :={x∈ X |γ(x) =minγ}is dense in X. Then the dynamical system{fⁿ}n∈ZZZ is topologically transitive.

We can show Lemma 2.4, Lemma 2.5, Lemma 2.6 in the same way as Lemma 2.1, Lemma 2.2, Lemma 2.3.

Proof of Theorem 2.4. By Lemma 2.4, (1) implies (2) and the inclusionD₊⊂M_γ holds.

By Lemma 2.6, (2) implies (1). Furthermore, sinceκ ∈ΓandM_κ=D, the inclusion

D⊂¹_γ∈ΓM_γ holds.

3 Sensitive dependence on initial conditions in dynamical sys- tems

(1) Sensitive dependence on initial conditions with the parameter NNN^∗ = NN

N∪ {0}

Let (X,d) be a metric space. Sensitive dependence on initial conditions is the property characterized by the metricd. In Theorem 3.1 we show that the constantδ>0 in the definition can be taken to be dependent onx∈X. First we introduce the definition of sensitive dependence on initial conditions.

Definition 3.1. Let (X,d) be a metric space, f a continuous map from X to X, and {fⁿ}n∈NNN^∗ the dynamical system defined by the continuous map f. The dynamical sys- tem{fⁿ}_n∈NNN^∗ is said to havesensitive dependence on initial conditionsif there exists a constantδ >0 such that for anyx∈X and any open neighborhoodUofxwithU≥2, there exist a pointy∈Uand a nonnegative integern≥0 satisfyingd(fⁿ(x),fⁿ(y))>δ. In what follows, we use the notationa∧banda∨bto denote min{a,b}and max{a,b} respectively.

Definition 3.2. Forx∈X and anε>0, the numbersr₊(ε,x)andr₊(x)are defined as follows:

r₊(ε,x) =

def sup

n≥0 sup

y,y∈Bε(x)d(fⁿ(y),dⁿ(y))∧1 r₊(x) =

def lim

ε↓0r₊(ε,x)

= lim

ε↓0

sup

n≥0

sup

y,y∈B_ε(x)d(fⁿ(y),dⁿ(y))∧1

,

whereB_ε(x)is the open ball atx∈Xwith radiusε>0.

Remark. Ifx∈Xis an isolated point, there is anε0>0 such thatB_ε0(x) ={x}. Hence r₊(ε,x) =0 for any 0<ε≤ε0, sor₊(x) =0.

Theorem 3.1. Let(X,d)be a metric space and f a continuous map from X to X. As- sume that the dynamical system{fⁿ}_n∈NNN^∗is topologically transitive. Then the following two conditions are mutually equivalent.

(1) The dynamical system{fⁿ}_n∈NNN^∗ has sensitive dependence on initial conditions.

(2) There existsδ =δ(x)>0for any x∈X satisfying the following conditions: for any open neighborhood U of x withU≥2, there are a nonnegative integer n≥0 and a point y∈U such that d(fⁿ(x),fⁿ(y))>δ(x).

To prove this theorem, we give some Lemmas: Lemma 3.1, Lemma 3.2. Lemma 3.1 and Lemma 3.2 show thatr₊ is the function which characterizes sensitive dependence on initial conditions.

Lemma 3.1. Let(X,d)be a metric space and f a continuous map from X to X. Then r₊is upper semicontinuous.

Proof. For any givenx0∈X and anyε >0, there existsε0>0 such thatr₊(ε0,x0)≤ r₊(x₀)+ε. For anyx∈B_ε0(x₀)there is anε1>0 satisfyingB_ε1(x)⊂B_ε0(x₀), hence,r₊

is upper semicontinuous.

Lamma 3.2. Let(X,d)be a metric space and f a continuous map from X to X. Then r₊is f -subinvariant. Furthermore, if f is an open map, r₊is f -invariant.

Proof. For anyx∈X and anyε>0, f⁻¹(B_ε(f(x)))is an open neighborhood ofx. So there exists an ε1>0 satisfying B_ε1(x)⊂ f⁻¹(B_ε(f(x))). The following inequalities hold for anyε>0 with 0<ε≤ε1.

r₊(x) ≤ r₊(ε,x)

≤ 2ε∨

sup

n≥1

sup

y,y∈B_ε(x)d(fⁿ(y),fⁿ(y))∧1

≤ 2ε∨

sup

n≥1

sup

y,y∈f⁻¹(B_ε(f(x)))

d(fⁿ(y),fⁿ(y))∧1

≤ 2ε∨

supn≥0 sup

y,y∈B_ε(f(x))d(fⁿ(y),fⁿ(y))∧1

= 2ε∨r₊(ε,f(x)).

Letting ε ↓ 0, we have r₊(x)≤r₊(ε,f(x)). Next letting ε ↓0, we have r₊(x) ≤ r₊(f(x)). Thusr₊ is f-subinvariant. Suppose that f is an open map. Then, f(B_ε(x))is an open neighborhood of f(x)for anyx∈X andε>0. Since the relation

sup

y,y∈f(B_ε(x))

d(fⁿ(y),fⁿ(y))∧1= sup

z,z∈B_ε(x)

d(fⁿ⁺¹(z),fⁿ⁺¹(z))∧1 holds for arbitraryn≥0, we have

sup

n≥0

sup

y,y∈f(B_ε(x))d(fⁿ(y),fⁿ(y))∧1 = sup

n≥1

sup

z,z∈B_ε(x)d(fⁿ(z),fⁿ(z))∧1

≤ sup

n≥0

sup

z,z∈B_ε(x)d(fⁿ(z),fⁿ(z))∧1. Therefore

r₊(f(x)) ≤ sup

n≥0 sup

y,y∈f(Bε(x))d(fⁿ(y),fⁿ(y))∧1

≤ r₊(ε,x).

Letε ↓0. Thenr₊is f-invariant.

Proof of Theorem 3.1. From triangle inequality, the inequalities sup

y∈Bε(x)d(fⁿ(x),fⁿ(y))∧1 ≤ sup

y,y∈Bε(x)d(fⁿ(y),fⁿ(y))∧1

≤ 2 sup

y∈Bε(x)d(fⁿ(x),fⁿ(y))∧1

hold for anyn≥0. From these inequalities, we can see that the conditions (1) and (2) of Theorem 3.1 are equivalent to the following(1)and(2)respectively.

(1) There exists a constant δ >0 satisfying r₊(ε,x)>δ for any x∈X and any ε >0.

(2) For anyx∈X, there exists a constantδ(x)>0 satisfyingr₊(ε,x)>δ(x)for any ε >0.

Furthermore, the conditions(1)and(2)are equivalent to the following(1)and(2) respectively.

(1) There exists aδ>0 satisfyingr₊(x)>δfor anyx∈X.

(2) r₊(x)>0 for anyx∈X.

Lemma 3.1 and Lemma 3.2 imply thatr₊is upper semicontinuous and f-subinvariant and hencer₊∈Γs. Sincer₊has a minimum from Lemma 2.1, we see that(2)implies

(1). The converse is trivial.

(2) Sensitive dependence on initial conditions with the parameterZZZ

We discuss sensitive dependence on initial conditions when the parameter of the dynamical system{fⁿ}n∈ZZZruns overZZZ. Theorem 3.2 below says that Theorem 3.1 also holds in the case when f is a homeomorphism. First we introduce the definition of sensitive dependence on initial conditions in the case when the parameter runs overZZZ.

Definition 3.3. Let(X,d) be a metric space, f a homeomorphism fromX toX, and {fⁿ}n∈ZZZ the dynamical system defined by f. The dynamical system{fⁿ}n∈ZZZis said to havesensitive dependence on initial conditionsif there exists a constantδ>0 such that for anyx∈X and any open neighborhoodU ofxwithU≥2, there exist a pointy∈U and an integern∈ZZZsatisfyingd(fⁿ(x),fⁿ(y))>δ.

Definition 3.4. For x∈X and anε >0, the numbers r(ε,x) andr(x)are defined as

follows:

r(ε,x) =

def sup

n∈ZZZ

sup

y,y∈B_ε(x)d(fⁿ(y),fⁿ(y))∧1, r(x) =

def lim

ε↓0r(ε,x)

= lim

ε↓0

sup

n∈ZZZ

sup

y,y∈B_ε(x)d(fⁿ(y),fⁿ(y))∧1

.

Theorem 3.2. Let(X,d) be a metric space, f a homeomorphism from X to X, and {fⁿ}n∈ZZZ the dynamical system defined by f . Suppose that {fⁿ}n∈ZZZ is topologically transitive. Then the following two conditions are mutually equivalent.

(1) The dynamical system{fⁿ}n∈ZZZhas sensitive dependence on initial conditions.

(2) For any x∈X there exists a constantδ =δ(x)>0satisfying the following condi- tions: for any open neighborhood U of x withU≥2, there are an integer n∈ZZZ and a point y∈U such that d(fⁿ(x),fⁿ(y))>δ(x).

To show Theorem 3.2, we will give some Lemmas: Lemma 3.3, Lemma 3.4.

Lemma 3.3. Let(X,d)be a metric space and f a homeomorphism from X to X. Then r(x)is upper semicontinuous.

Lemma 3.3 is shown in the same way as Lemma 3.1.

Lemma 3.4. Let(X,d)be a metric space and f a homeomorphism from X to X. Then r(x)is f -invariant.

Proof. Since f is a homeomorphism, f⁻¹(B_ε(f(x)))is an open neighborhood ofxfor anyx∈Xand anε>0. There exists anε1>0 satisfyingB_ε1(x)⊂f⁻¹(B_ε(f(x))). Then

r(x) ≤ r(ε1,x)

≤ sup

n∈ZZZ

sup

y,y∈f⁻¹(Bε(f(x)))d(fⁿ(y),fⁿ(y))∧1

= sup

n∈ZZZ

sup

z,z∈Bε(f(x))d(fⁿ(y),fⁿ(y))∧1

= r(ε,f(x)).

Lettingε ↓0, we see thatr(x)is f-subinvariant. We show the converse inequality. For anyx∈X and anyε>0 given, f(B_ε(x))is an open neighborhood of f(x)and there is

ε>0 satisfyingB_ε(f(x))⊂ f(B_ε(x)). Then r(f(x)) ≤ r(ε,f(x))

≤ sup

n∈ZZZ

sup

y,y∈B_ε(f(B_ε(x)))d(fⁿ(y),fⁿ(y))∧1

= sup

n∈ZZZ

sup

z,z∈B_ε(x)d(fⁿ(f⁻¹(z)),fⁿ(f⁻¹(z)))∧1

= r(ε,x).

Hencer(x)is f-invariant.

Proof of Theorem 3.2. From triangle inequality, the inequalities sup

y∈B_ε(x)

d(fⁿ(x),fⁿ(y))∧1 ≤ sup

y,y∈B_ε(x)

d(fⁿ(y),fⁿ(y))∧1

≤ 2 sup

y∈B_ε(x)

d(fⁿ(x),fⁿ(y))∧1

hold forn∈ZZZ. By these inequalities, we see in the same way as in the proof of Theorem 3.1 that conditions (1) and (2) of Theorem 3.2 are equivalent to the following conditions (1)and(2)respectively.

(1) There is aδ>0 such thatr(x)>δfor anyx∈X.

(2) r(x)>0 for anyx∈X.

Lemma 3.3 and Lemma 3.4 yieldr∈Γ. Sincerhas a minimum from Lemma 2.4,(2)

implies(1). The converse is trivial.

4 Applications of new characterizations to topological transi- tivity

In thisfinal section, we apply the new characterization of topological transitivity in the previous sections to prove the main results.

Theorem 4.1. Let(X,d)be a complete separable metric space without isolated points and f a homeomorphism from X to X. Suppose that the dynamical system {fⁿ}_n∈ZZZ

defined by f is topologically transitive. Then both{fⁿ}_n∈NNN^∗ and{f⁻ⁿ}_n∈NNN^∗ are topo- logically transitive. Furthermore, D₊∩D₋is dense in X.

To prove this theorem, we need the following Lemma 4.1 and Lemma 4.2.

Lemma 4.1. Let(X,d)be a complete separable metric space without isolated points, f a homeomorphism from X to X, andγ+,γ−upper semicontinuous functions from X to [0,∞). Suppose that the dynamical system{fⁿ}n∈ZZZ is topologically transitive and that γ+ is f -subinvariant,γ− is f -superinvariant. Ifinfγ+=infγ−=0, thenγ+∧γ− has a minimum0.

Proof. inf(γ+∧γ−) =0 is trivial from infγ+=infγ−=0. Letl∈NNN. We set A_l :=

x∈X

γ+(x)∧γ−(x)≥inf(γ+∧γ−) +1 l

.

A_l is closed set. Suppose that γ+∧γ− does not have a minimum. It is clear that X is represented as a countable union of A_l. From Baireʼs category theorem, there are a nonempty open set O andl0∈NNN with O⊂A_l0. We now put A:=A_l0 anda= _l¹₀. Since the dynamical system{fⁿ}n∈ZZZis topologically transitive, Theorem 2.3 yields that D is dense inX. Letx₀∈D. SinceX has no isolated points, we can see that the set {n∈ZZZ| fⁿ(x0)∈A}is infinite by{n∈ZZZ| fⁿ(x0)∈O}=∞and the inclusionA⊃O.

When the set{n∈ZZZ | fⁿ(x₀)∈A}is unbounded from above, there is anm∈ {n∈ ZZZ| fⁿ(x₀)∈A}satisfyingn≤mfor any integern∈ZZZ. Thusγ−(fⁿ(x₀))≥aforn∈ZZZ.

a≤limγ−(y_n)≤γ−(y) holds for anyy∈X because of the upper semicontinuity of γ and the denseness ofO(f;x₀). Therefore we founda≤γ−(y)for anyy∈X. This is a contradiction to infγ−=0. When the set{n∈ZZZ|fⁿ(x₀)∈A}is unbounded from below, there is anm∈ {n∈ZZZ | fⁿ(x₀)∈A}satisfyingn≥mfor any integern∈ZZZ, hence, we can see a≤γ+(f^m(x₀))≤γ+(fⁿ(x₀)). We have thatγ+(x)≥aholds for anyx∈X in the same way. This contradicts to infγ+=0. Thereforeγ+∧γ−has a minimum 0.

Lemma 4.2. Let X be a complete separable metric space without isolated points and f a continuous map from X to X. Moreover letγ+,γ−be upper semicontinuous functions from X to [0,∞)such that γ+ is f -subinvariant and γ− is f -superinvariant. Suppose that the dynamical system {fⁿ}n∈NNN^∗ is topologically transitive. Then γ+∨γ− has a minimum. Furthermore, if f is a homeomorphism from X to X, thenmin(γ+∨γ−) = minγ+∨minγ−holds.

Proof. Puta:=inf(γ+∨γ−), and B_n:=

x∈X

γ+∨γ−(x)≥a+1 n

.

B_nis a closed set. Suppose thatγ+∨γ−does not have a minimum, thenXis represented as a countable union ofBn. From Baireʼs category theorem, there exists ann0∈NNNsuch thatB_n0 has an inner point. Thus there are a real numberb∈(a,∞) and a nonempty

open setOwithO⊂ {x∈X|γ+(x)∨γ−(x)≥b}. For anyx∈D₊there is a nonnegative number n(x)≥0, which depends on x, satisfying f^n(x)(x)∈Ofrom the definition of D₊. Hence we can see thatγ+(f^n(x)(x))≥borγ−(f^n(x)(x))≥b.

First, we consider the case when there exists a pointx∈D₊satisfyingγ+(f^n(x)(x))≥ b. Sinceγ+ is f-subinvariant,γ+(f^m(x))≥bform≥n(x). Since the set{f^m(x)|m≥ n(x)} is dense in X and the previous inequality, γ+(y)≥b for any y∈X. Thus we haveγ+∨γ−(y)≥bfor anyy∈X. This contradicts tob>a. Secondly, we consider the case whenγ−(f^n(x)(x))≥bholds for anyx∈D₊. Sinceγ−is f-superinvariant, the inequalities b≤γ−(f^n(x)(x))≤γ−(x)holds for any x∈D₊. Hence these inequalities and the definition ofD₊ lead to the fact that the inequality γ−(y)≥b holds for any y∈X. Thusγ+∨γ−(y)≥b for anyy∈X. This also contradicts tob>a. Therefore γ+∨γ− has a minimum.

We assume that f is a homeomorphism and puta:=min(γ+∨γ−). Since the dy- namical system {fⁿ}n∈NNN^∗ is topologically transitive, the dynamical system{f⁻ⁿ}n∈NNN^∗

is also topologically transitive. Thus,γ+ has a minimum from Lemma 2.1 andγ− also has a minimum sinceγ−is f⁻¹-subinvariant.

Puta₊:=minγ+ anda₋:=minγ−. Suppose thata>a₊∨a₋. First, we consider the case whena₊ ≥a₋. Clearlya>a₊. From Lemma 2.1, the inclusionD₊⊂ {x∈ X |γ+(x) =a₊}holds. Henceγ−(x)≥afor anyx∈D₊and therefore we can see that γ−(y)≥a for anyy∈X. This contradicts toa>a₊∨a₋. Secondly, when the case a₊<a₋holds, we have a contradiction in the same way.

We havea≤a₊∨a₋, the conversea≥a₊∨a₋ is trivial, hence,a=a₊∨a₋. Proof of Theorem 4.1. Let{U_n}be a countable basis ofX which is nonempty for any integern≥1. We define a setFm⁽ⁿ⁾and mapsκ+(x),κ−(x),κ(x) (x∈X)as follows.

F_m⁽ⁿ⁾ := f^−m(U_n^c) (m∈ZZZ, n≥1) κ+(x) :=

∑

n=1

1 3ⁿ·

∏

m=0

111Fm⁽ⁿ⁾(x) (x∈X) κ−(x) :=

∑

n=1

1

3ⁿ·_m=−∞

∏

∑

n=1

1

3ⁿ·_m

∏

κ−(f(x)) =

∑

n=1

1 3ⁿ·

∏

m=−∞

111Fm⁽ⁿ⁾(f(x))

≤

∑

n=1

1

3ⁿ·_m=−∞

∏