Omega chaos and the specification property

J. Math. Anal. Appl. 448 (2017) 908–913

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Omega chaos and the specification property Reeve Hunter a,∗ , Brian E. Raines b a b

Department of Mathematics, University of Georgia, Athens, GA 30602-5016, USA Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA

a r t i c l e

i n f o

Article history: Received 7 June 2016 Available online 21 November 2016 Submitted by J. Bonet Keywords: Chaotic system Periodic point Shift map Symbolic dynamics Specification property Omega chaos

a b s t r a c t In this short paper we consider the connections between the specification property introduced by Bowen and ω-chaos introduced by Li. We show that if f : X → X is a continuous surjection of a compact metric space with the specification property and uniform expansion near a fixed point then the system (X, f ) is ω-chaotic. Published by Elsevier Inc.

1. Introduction and preliminaries In 1971, R. Bowen introduced the specification property for a map on a compact metric space, [2]. We say that f : X → X has the specification property provided for every δ > 0 there is some Nδ ∈ N such that for n ≥ 2 and for any n points x1 , . . . , xn ∈ X and any sequence of natural numbers a1 ≤ b1 < a2 ≤ b2 < · · · < an ≤ bn with ai − bi−1 ≥ Nδ for 2 ≤ i ≤ n there is a periodic point x ∈ X such that d(f j (x), f j (xi )) < δ for ai ≤ j ≤ bi and 1 ≤ i ≤ n. The map has the weak specification property if the above holds with n = 2. This is a strong property for a dynamical system. If the map is in addition expansive, Bowen showed that there is a unique equilibrium measure for any smooth potential, [3]. For an explanation of this result along with some other early results involving the specification property, see [6]. Despite the apparent strength of this property, Hofbauer showed that all continuous piecewise monotone maps of the interval have the weak specification property, [7], and Blokh showed that maps of the interval have the specification property if, and only if, they are topologically mixing, [1]. More recently, Buzzi has characterized the specification property in the context of piecewise continuous, piecewise monotone maps of the interval, [5]. In 2009, Lampart and Oprocha considered the weak specification property on symbolic dynamical systems, [8]. They characterized this property and the specification property on shift spaces. They also show * Corresponding author. E-mail addresses: [email protected] (R. Hunter), [email protected] (B.E. Raines). http://dx.doi.org/10.1016/j.jmaa.2016.11.037 0022-247X/Published by Elsevier Inc.

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that a shift space satisfying the weak specification property has a form of chaos called ω-chaos, introduced by S.H. Li in [9]. Let f : X → X be a map, and let z ∈ X. The ω-limit set of z is the set of all limit points of the orbit of z under f , and it is denoted by ωf (z) or simply ω(z). We say that f has ω-chaos if there is an uncountable collection Ω of points of X such that for any x, y ∈ Ω with x = y we have (i) ω(x) \ ω(y) is uncountable, (ii) ω(x) ∩ ω(y) = ∅, (iii) ω(x) is not made up of only periodic points. In this paper we partially answer a question raised by Lampart and Oprocha, [8], by showing that continuous surjections on compact metric spaces with the specification property and uniform expansion near a fixed point have ω-chaos. Our main theorem, specifically, is that if f : X → X is a continuous surjection with the specification property, X is compact metric, and there is some fixed point s ∈ X, η > 0 and λ > 1 such that if d(s, y) < η then d(s, f (y)) ≥ λd(s, y), then the system (X, f ) has ω-chaos. We show this by encoding subsystems of (X, f ) with subshifts from (2ω , σ), the full shift on two symbols. We now provide a few of the basic definitions and results needed throughout the paper. The interested reader should see [4] or [10] for additional background. In the following section we carefully construct some subsystems of (X, f ) via an encoding of subshifts of (2ω , σ) and prove the main theorem. We use ω to stand for the set of natural numbers with 0. Let A = {0, 1} have the discrete topology,  and let 2ω = n∈ω A have the product topology. We call 2ω the full shift on 2 symbols. Let the shift map σ : 2ω → 2ω be defined by σ(x0 , x1 . . . ) = (x1 , x2 . . . ). If K ⊆ 2ω is closed and σ-invariant, i.e. σ(K) = K, then we call K a subshift of 2ω . Given α ∈ 2ω and k ∈ ω we let α[0,k+1) = α0 , α1 , . . . , αk . Let f : X → X be a continuous map on a compact metric space (X, d). Let K ⊆ X. We call K f -invariant if f (K) = K. We say that K is minimal provided K is closed and f -invariant and there is no closed proper subset of K that is also f -invariant. Let x ∈ X. We say that x is uniformly recurrent provided for every ε > 0 there is some Mε such that if f j (x) ∈ Bε (x), for j ≥ 0, then there is some 1 ≤ k ≤ Mε such that f j+k (x) ∈ Bε (x). It is a well known fact that K is minimal if, and only if, every point x of K is uniformly recurrent, [4]; moreover if x is uniformly recurrent then ω(x) is minimal. It is not hard to construct an uncountable collection of points Λ ⊆ 2ω that are uniformly recurrent with the property that if x, y ∈ Λ and x = y then ω(x) ∩ ω(y) = ∅ and each ω(x) is uncountable. Thus there are uncountably many disjoint minimal subsets of 2ω . We will use these points in what follows to find an uncountable collection of points witnessing ω-chaos. 2. Constructions and proof of the main theorem In order to show f is ω-chaotic, we will exhibit an uncountable omega scrambled set. The construction of this set will rely on the fact mentioned above that 2ω has an uncountable set of points whose ω-limit sets form an uncountable collection of minimal sets. We will prove our main theorem via a careful encoding of points from 2ω into our dynamical system f : X → X with the specification property and uniform expansion near a fixed point. We begin by fixing periodic points t0 , t1 ∈ X. Then, for each α ∈ 2ω , we will construct a point xα ∈ X whose iterates under f get close to t0 or t1 in a pattern determined by the coordinates of α. The point xα will also, under iterations of the map f , get close to the fixed point s mentioned in the theorem.

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For the arguments and lemmas that follow, we will assume f : X → X is a continuous surjection with the specification property on a compact metric space. In addition, assume that there is a fixed point s such that there is some η > 0 and λ > 1 such that if 0 < d(s, y) < η then d(s, f (y)) ≥ λd(s, y). Since f has the specification property, there are points t0 , t1 ∈ X periodic with periods p and q, respectively, and t0 = t1 , t0 = s = t1 . Let > 0 such that {B2 (s), B2 (f i (t0 )), B2 (f j (t1 ))} 0≤i
0≤j
is a disjoint collection of sets. Since f is uniformly continuous, choose δ  > 0 such that if d(y, z) < δ  , then d(f i (y), f i (z)) < , for all 0 ≤ i ≤ pq. Let 0 < δ < min{η, , δ  }. Let N witness specification property for δ. Choose α ∈ 2ω . We will consider the following sequence of points in X: Φ(α) = (tα0 , s, tα0 , tα1 , s, tα0 , tα1 , tα2 , s . . .) where the indices on the α’s follow (0, 0, 1, 0, 1, 2, 0, 1, 2, 3 . . .). Notice that any initial segment of α is repeated infinitely often as indices. We will refer to Φ(α) as the pattern induced by α and denote the jth term of Φ(α) by Φ(α)j . Next, using the specification property we will construct a point xα ∈ X such that xα ’s iterates follow the pattern of Φ(α) in the correct order. The iterates of xα will stay within δ of iterates of each tαi for 2N iterates. For the first s that appears in Φ(α), we want the iterates of xα to remain within δ of s for 4N many iterations. The kth instance of s in Φ(α) corresponds to iterates of xα staying within δ of s for 4k N many consecutive iterations. To this end, we recursively construct a sequence of integers 0 = p 1 < q1 < p 2 < q2 < p 3 < q3 < · · · , so that p1 = 0

q1 = 2N

iterates of xα are near tα0

p 2 = q1 + N

q2 = p2 + 4N

iterates of xα are near s

p 3 = q2 + N

q3 = p3 + 2N

iterates of xα are near tα0

p 4 = q3 + N

q4 = p4 + 2N

iterates of xα are near tα1

p 5 = q4 + N

2

q5 = p5 + 4 N

.. .

iterates of xα are near s 2nd time

.. .

Continuing in this manner, we define pi and qi for all i ∈ N. Note the following association between terms of Φ(α) and each pj for j ∈ N. Φ(α) = tα0 p1

s p2

t α0 p3

t α1 p4

s p5

t α0 p6

t α1 p7

t α2 p8

s p9

... ...

We are now ready to define xα . Lemma 1. For each α ∈ 2ω , there is a point xα ∈ X such that   d f i (xα ) , f i−pj (Φ(α)j ) ≤ δ, pj ≤ i < qj , j ∈ ω.

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Proof. Note that since s is a fixed point, we have that f i (s) = s, for all i ∈ ω. Then, by the specification property, we have the following: There is x1α such that     d f i x1α , f i (tα0 ) < δ,

p1 ≤ i < q1 .

There is x2α such that     d f i x2α , f i (tα0 ) < δ,     d f i x2α , s < δ,

p1 ≤ i < q1 and p2 ≤ i < q2 .

There is x3α such that     d f i x3α , f i (tα0 ) < δ,     d f i x3α , s < δ,     d f i x3α , f i−p3 (tα0 ) < δ,

p1 ≤ i < q1 , p2 ≤ i < q2 , and p3 ≤ i < q3 .

Generally, there is xkα such that     d f i xkα , f i−pj (Φ(α)j ) < δ,

pj ≤ i < qj ,

for j ≤ k.   Then xkα k∈N is a sequence of points in Bδ (tα0 ). Denote a limit point of this sequence xα and, for each α ∈ 2ω , note that we may define such an xα . Then xα has the desired property. 2 Note that in the above proof we use the fact that f is a surjective mapping to make sure iterates of xkα fall within δ of Φ(α)j rather than some iterate of Φ(α)j . Similar strategy is used elsewhere in this paper. In order to prove the main result, we need to prove a few simple lemmas regarding the points we constructed in the previous lemma and points in their ω-limit set. Lemma 2. For any α ∈ 2ω , s ∈ ωf (xα ). Proof. Let α ∈ 2ω and xα as in Lemma 1. Note that λd(s, y) ≤ d(s, f (y)) whenever d(s, y) < η. Suppose λn d(s, y) ≤ d(s, f n (y)) with d(s, f n (y)) < η. Then λn+1 d(s, y) ≤ λd(s, f n (y)) ≤ d(s, f n+1 (y)). Suppose for all 0 ≤ i ≤ n we have d(s, f i (y)) < η. Then λn d(s, y) ≤ d(s, f n (y)) < η d(s, y) ≤

1 η d(s, f n (y)) < n . n λ λ

Let 0 < ν < η and let m ∈ N such that ληm < ν. Then if d(s, f i (y)) ≤ δ < η, for all i ≤ m, d(s, y) < η i λm < ν. Let n ∈ N such that qn − pn ≥ m, 2N . Then d(f (xα ), s) ≤ δ < η for pn ≤ i < qn and so pn d(f (xα ), s) < ν. Since this holds for any ν with 0 < ν < η, s ∈ ωf (xα ). 2 Next, we will choose β ∈ ωσ (α). We will then define a point zβ whose iterates get close to iterates of t0 and t1 in a pattern determined by the coordinates of β—we do not mean Φ(β) here, but rather a pattern considering only t0 and t1 while not including the fixed point s. To this end, let c0 = 0. Then define di = ci + 2N and ci+1 = di + N for i ∈ ω. So we have

912

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c0 = 0

d0 = 2N

c1 = 3N

d1 = 5N

c2 = 6N

d2 = 8N

c3 = 9N

d3 = 11N

.. .

.. .

Lemma 3. Suppose α ∈ 2ω with β ∈ ωσ (α). Then there is a point zβ ∈ ωf (xα ) such that d(f i (zβ ), f i−cj (tβj )) ≤ δ for all cj ≤ i < dj for all j ∈ ω. Proof. Let α ∈ 2ω and let β ∈ ωσ (α). Then there is some increasing sequence of natural numbers (rn )n∈ω such that σ rn (α)[0,n+1) = β[0,n+1) for all n ∈ ω. Note that if qk − pk ≥ 4r N from some k, r ∈ N, then Φ(α)r = s and Φ(α)r+j = s for 0 ≤ j ≤ r + 1. Let k0 ∈ N such that qk0 −1 − pk0 −1 ≥ 4r0 N . Suppose kn−1 is defined. Let kn ∈ N such that kn > kn−1 and qkn−1 − pkn−1 ≥ 4rn +n N .   Note that for all n ∈ ω, Φ(α)kn +rn +j = tβj for 0 ≤ j ≤ n. Letting in = pkn +rn , consider f in (xα ) n∈ω .     Noting that d f in (xα ), tβ0 ≤ δ for all n, let zβ be a limit point of f in (xα ) n∈ω . Then zβ has the property  i  that d f (zβ ), f i−cj (tβj ) ≤ δ for all cj ≤ i < dj for all j ∈ ω. 2 So for any β ∈ ωσ (α), we may define zβ ∈ ωf (xα ) ⊆ X such that zβ follows iterates of t0 and t1 according to the coordinates of β. We now show a converse of sorts. We show that if zβ also belongs to ωf (xγ ), then β ∈ ωσ (γ). Lemma 4. Suppose α, γ ∈ 2ω and β ∈ ωσ (α). Suppose further that zβ ∈ ωf (xα ) as in Lemma 3. Then, if zβ ∈ ωf (xγ ), β ∈ ωσ (γ). Proof. Recall that we chose > 0 so that {B2 (s), B2 (f i (t0 )), B2 (f j (t1 ))} 0≤i
0≤j
where p and q are the periods of t0 and t1 , respectively, is a disjoint collection of sets. Let U0 = ∪0≤i

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Using the previous lemmas, we are now ready to prove the main theorem. Theorem 5. Assume that X is a compact metric space with f : X → X a continuous surjective map with the specification property. Assume s ∈ X is a fixed point such that there is η > 0, λ > 1 such that if 0 < d(s, y) < η, then d(f (s), f (y)) ≥ λd(s, y). Then f is ω-chaotic. Proof. Let A ⊂ 2ω such that {ωσ (α)}α∈A is an uncountable family of uncountable minimal sets with ωσ (α) ∩ ωσ (γ) = ∅, for all α, γ ∈ A, α = γ. Then for each α ∈ A we may define a point xα as in Lemma 1. By Lemmas 3 and 4, for each β ∈ ωσ (α), there is a point zβ ∈ ωf (xα ) such that zβ ∈ / ωf (xγ ) for all γ ∈ A − {α}. Note that the collection {zβ : β ∈ ωσ (α)} is uncountable. Note, too, that since ωσ (α) is minimal, β ∈ ωσ (α) is not periodic. Hence, by construction, zβ is not periodic for all β ∈ ωσ (α). Let Ω = {xα }α∈A . Note that for any xα , xγ ∈ Ω, we have (ωf (xα ) \ ωf (xγ )) is uncountable. Note, too, that ωf (xα ) ∩ ωf (xγ ) s by Lemma 2. Finally, we have that ωf (xα ) \ Per(f ) = ∅. Hence f is ω-chaotic. 2 It is an unknown if a weaker form of the specification property is enough to obtain a result similar to the main theorem presented in this paper. In addition, it would be interesting to know if there is a way to drop the assumption of an expanding fixed point and still achieve the same result. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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