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Minimality and Gluing Orbit Property

We show that a topological dynamical system is either minimal or have positive topological entropy. Moreover, for equicontinuous systems, we show that topological transitivity, minimality and orbit gluing property are equivalent. These facts reflect the similarity and dissimilarity of gluing orbit property with specification property.


Introduction
The notion of gluing orbit property was introduced in [12], [5] and [2]. As a weaker form of the well-studied specification properties, it turns out to be a more general property which still captures crucial topological features of the systems, especially the non-hyperbolic ones. A number of results have been obtained based on this property. See also [1], [4], [13] and [14]. For classical results with specification property and specification like properties, the readers are referred to [6] and [10].
There is a remarkable difference between gluing orbit property and specification property, as well as most weaker forms of the latter. As illustrated in [1] and [2], certain examples far from specification, such as irrational rotations, have gluing orbit property. We can see that gluing orbit property only requires topological transitivity, and is compatible with zero topological entropy, while specification property implies topological mixing and positive topological entropy. In general, topological mixing should not be expected for a system that only has gluing orbit property. For example, the direct product of the irrational rotation and any system with specification property has gluing orbit property and is not topologically mixing. In this article, we consider the entropy and find that there is a dichotomy: a system with gluing orbit property is either minimal or of positive topological entropy.
As a direct corollary of Theorem 1.1, periodic gluing orbit property implies positive topological entropy, just as the specification properties do. The only exception that should be ruled out is the trivial case that X consists of a single periodic orbit. Corollary 1.2. A non-trivial system with the periodic gluing orbit property must have positive topological entropy and exponential growth of periodic orbits.
By Theorem 1.1, for a system with zero topological entropy, gluing orbit property implies minimality. Example 6.6 shows that the converse is not true. We can show that the converse holds if the the system is equicontinuous, in which case gluing orbit property is also equivalent to topological transitivity. This extends the examples in [1]. We doubt if there are systems that are not equicontinuous, of zero topological entropy and have gluing orbit property (hence minimal). Theorem 1.3. Assume that (X, f ) is equicontinuous. Then the followings are equivalent: Our results hold for both invertible and non-invertible cases, and both discretetime and continuous-time cases as well. In this article we mainly work with homeomorphisms. There are some extra technical difficulties in the proof of the noninvertible and continuous-time cases. We give a proof of Theorem 1.1 in the semiflow case in Section 7 to illustrate the difference.
Some preliminaries are introduced in Section 2, including definitions and notations we shall use. We prove Theorem 1.1 in Section 3 and discuss some corollaries in Section 4. Theorem 1.3 is proved in Section 5. Some examples are investigated in Section 6.

Preliminaries
Let (X, d) be a compact metric space. Let f : X → X be a homeomorphism on X. Conventionally, (X, f ) is called a topological dynamical system or just a system. Definition 2.1. (X, f ) is said to be equicontinuous if for every ε > 0, there is δ > 0 such that for any x, y ∈ X with d(x, y) < δ, we have d(f n (x), f n (y)) < ε for every n ≥ 0.
Definition 2.4. For n ∈ Z + and ε > 0, a subset E ⊂ X is called an (n, ε)-separated if for any distinct points x, y in E, there is k ∈ {0, · · · , n − 1} such that Denote by s(n, ε) the maximal cardinality of (n, ε)-separated subsets of X. Then the topological entropy of f is defined as lim sup n→∞ ln s(n, ε) n .
Definition 2.5. We call the finite sequence of ordered pairs an orbit sequence of rank k. A gap for an orbit sequence of rank k is a (k − 1)-tuple For ε > 0, we say that (C , G ) can be ε-shadowed by z ∈ X if for every j = 1, · · · , k, (X, f ) is said to have periodic gluing orbit property if for every ε > 0, there is M (ε) > 0 such that for any orbit sequence C , there are t ≤ M (ε) and a gap G with max G ≤ M (ε) such that (C , G ) can be ε-shadowed by a periodic point of the period s k + m k + t.
The notion of specification property was first introduced by Bowen in [3]. It has a number of variations and their names also vary in different literatures. An overview of these specification like properties can be found in [10]. Gluing orbit property first appeared in [12], where it is called transitive specification. It is called weak specification in [5] in a slightly generalized form. It is in [2] that the name gluing orbit is called to indicate its dissimilarity with specification like properties.
Here we attempt to reformulate the definitions of specification and gluing orbit properties to make our argument more clear and more convenient. We follow the names called in [2], [10] and [13]. Note that in our definitions of periodic specification and periodic gluing orbit properties the gap G may be ∅.
Definition 2.5 naturally extends to infinite orbit sequences. Definition 2.6 and 2.8 are conventional definitions speaking of finite orbit sequences. However, they are equivalent to the definitions speaking of infinite ones. This is clear for specification. For gluing orbit property a little extra work should be done. The flow version of the following lemma is contained in [4]. A similar technique is also part of the proof of Theorem 1.1.

Lemma 2.10. (X, f ) has gluing orbit property if and only if for every
Proof. The if part is trivial.
Apply this procedure inductively, we obtain a sequence G = {t j } ∞ j=1 and subsequence {z n(j,k) } for each j ∈ Z + such that t j (n(j, k)) = t j for every k.
Let z be a subsequential limit of {z n(k,k) }. Then for every j ∈ Z + and l = 0, 1, · · · , m j − 1, Initial idea of the proof of Theorem 1.1 comes from the following classical result. Proposition 2.11 (cf. [3]). Assume that (X, f ) has specification property. Assume that ε > 0 and there is a subset E of X that is (1, 3ε)-separated and |E| = N ≥ 2. Then and hence

This implies that
is an (mn, ε)-separated set and hence s(mn, ε) ≥ |A n | = N n . This yields that Corollary 2.12. Assume that (X, f ) has specification property and X is not a singleton. Then the followings hold.

Positive Entropy
Recall that a point Given a non-minimal system with gluing orbit property, to show that it has positive topological entropy, our idea is based on existence of two non-recurrent points such that the forward orbit of one point stays away from the other point, and vice versa.
Note that without gluing orbit property, a non-minimal system may have no nonrecurrent points and a system with non-recurrent points may have zero topological entropy (cf. Example 6.3 and 6.4).
Lemma 3.1. Assume that (X, f ) is not minimal and has gluing orbit property. Then f has a non-recurrent point.
Proof. As f is not minimal, there is a point whose orbit is not dense. We can find x, y ∈ X and δ > 0 such that We can find a subsequence {z ′ n k } with G n k = G for every k. Let z n k = f n k +t1−1 (z ′ n k ) and z be a subsequential limit of {z n k }. Then This also indicates that z is not periodic. So we have Then z is a non-recurrent point as is ε 1 -shadowed by y ′ n . Let y n = f tn (y ′ n ). Then d(f j (y n ), f j (x)) < ε 1 for j = 0, 1, · · · , n − 1.
As t n ∈ {1, · · · , m 1 } for every n, there is t such that {t n : t n = t} is infinite.
We can find a subsequence {y n k } such that t n k = t for every k.
Let y a a subsequential limit of {y n k }. Then which guarantees that y = x. Let Then for every n > 0, Now we complete the proof of Theorem 1.1. Let x, y ∈ X and ε > 0 be as in Lemma 3.2. Let ε 2 = 1 3 ε and m = M (ε 2 ). For each ξ = {x k (ξ)} n k=1 ∈ {x, y} n , consider Above argument shows that E = {z ξ : ξ ∈ {x, y} n } is a (2mn, ε 2 )-separated subset of X that contains 2 n points. Hence Proof. Assume that (X, f ) is not minimal and it has gluing orbit property. There are x, y ∈ X and δ > 0 such that d(f n (x), y) ≥ δ for every n ∈ Z.
By Lemma 2.10, there is y 0 ∈ X that ε-shadows (C , G ) for some G with max G ≤ m.
We have lim sup Denote by P n (f ) the set of periodic points of f with periods no more than n, and p n (f ) the cardinality of P n (f ). Consider A flow version of the following theorem is contained in [1].
For every x ∈ E, there is t < m such that {(x, n), ∅} is ε 2 -shadowed by a periodic point with period n + t. Hence every (n, ε 2 )-ball around an element of E, which are disjoint with each other, contains an element of P n+m (f ). This implies that It follows that The result follows as this holds for any h < h(f ).

Corollary 4.3.
Assume that (X, f ) has periodic gluing orbit property and X does not consist of a single periodic orbit. Then

Equicontinuous Systems
Let (X, f ) be an equicontinuous system . We shall show that minimality implies gluing orbit property. It is clear that gluing orbit property implies topological transitivity. For completeness, we present a proof that topological transitivity implies minimality. As every equicontinuous system has zero topological entropy, the fact that gluing orbit property implies minimality is also a corollary of Theorem 1.1.
We first prove a lemma that shows that the time needed for the pre-images of ε-balls to cover X is uniform. We remark that this lemma does not require equicontinuity.
Then d(f ny (z), x) ≤ d(f ny (y), x) + d(f ny (z), f ny (y)) < ε. This implies that As r is upper semi-continuous and X is compact, r attains its maximum R x < ε for every y ∈ X. This implies that Note that {B(x, δ x ) : x ∈ X} is an open cover of X. It has a finite subcover {B(x j , δ xj ) : j = 1, · · · , k}. Let N = max{N xj : j = 1, · · · , k}. Then for every x ∈ X, x ∈ B(x j , δ xj ) for some j and hence (B(x, ε)).
The proof of Theorem 1.3 is completed by Proposition 5.2 and Proposition 5.3.

Proposition 5.3. A topological transitive equicontinuous system is minimal.
Proof. Let x, y ∈ X and ε > 0. As f is equicontinuous, there is δ > 0 such that As f is topologically transitive, there is n ≥ 0 such that Then This implies that the orbit of every x ∈ X is dense, i.e. f is minimal.

Examples
Example 6.1. In [2], it is shown that a topologically transitive subshift of finite type has gluing orbit property. Note that such a system has periodic points. As a corollary of Theorem 1.1, it has positive topological entropy if it does not consist of a single periodic orbit. Example 6.5. According to [11], there are C ∞ interval maps with periodic points of period 2 n for any n ∈ Z + and zero topological entropy that are chaotic in the sense of Li-York. Theorem 1.1 implies that all such maps can not have gluing orbit property.
Example 6.6. The subshift on the closure of the orbit of an almost periodic point, as constructed in [7, 12.28] and [8], does not have gluing orbit property. The gap needed before shadowing an orbit segment of length L may be no less than L and hence neither uniform nor tempered. Such a system is minimal. It can have zero topological entropy and can also have positive topological entropy. So minimality itself does not imply gluing orbit property, no matter how much is the topological entropy (cf. Theorem 1.3).

The Semiflow Case
In this section we give a proof of Theorem 1.1 in the semiflow case. Throughout this section, f t is assumed to be a semiflow on X that is not minimal and has gluing orbit property. We first state the definition of gluing orbit property in this case and note the difference. Idea of the proof is similar to the homeomorphism case. There are two major technical differences. Non-recurrence is established after a time period and the orbit sequences for finding separated sets are more carefully designed.
Definition 7.1. A semiflow (X, f ) is said to have gluing orbit property if for every ε > 0 there is M (ε) > 0 such that for any orbit sequence there is a gap G = {t j ∈ [0, ∞) : j = 1, · · · , k − 1} such that max G ≤ M (ε) and (C , G ) can be ε-shadowed in the following sense: for every j = 1, · · · , k, There is x 0 ∈ X, ε > 0 and τ > 0 such that Proof. As f is not minimal, there is a point whose orbit is not dense. We can find x, y ∈ X and δ > 0 such that Let 0 < ε < 1 3 δ and m = M (ε). For each n ∈ Z + , consider There is τ n ∈ [0, m] such that (C n , {τ n }) is ε-shadowed by z n . There must be a subsequence {τ n k } that converges to τ ∈ [0, m]. Let x 0 be a subsequential limit of {z n k }. Then and There are x, y ∈ X, ε > 0 and T > 0 such that d(f t (y), y) ≥ ε for any t ≥ T, and d(f t (x), y) ≥ ε for any t ≥ 0.