ON SPATIOTEMPORAL CHAOS

Sergiǐ Kolyada

Abstract. The aim of this paper is to investigate the connection between various definitions of chaos for topological dynamical systems (i.e., continuous surjective maps of infinite compact metric spaces without isolated points into itself). Particular attention is paid to a very recent definition of spatiotemporal chaos which is based on Li–Yorke pairs and has some common features with sensitivity. We show that all topologically mixing systems, weakly mixing minimal systems, proximal systems and also some classes of recurrent systems are spatiotemporally chaotic.

0. Introduction and Definitions

The term ‘chaos’ in connection with a map was introduced by Li and Yorke [LY], although without a formal definition. Today there are various definitions of what it means for a map to be chaotic, some of them working reasonably only in particular phase spaces. Although the present article mainly deals with one of these definitions which was introduced in [BGKM], namely spatiotemporal chaos or simly ST chaos, it is also a survey on various definitions of chaos. Our main purpose is to incorporate the notion of ST chaos into the general frame of Topological Dynamics.

Let (X, T) be a topological dynamical system, with X a compact metric space with metric ϱ, and T a surjective continuous map from X to itself. Since topological transitivity is usually either a part of the definition of a chaos or transitivity implies or is implied by the chaos (at least in some spaces), we mention here some classes of transitive systems, some kinds of chaos and indicate their connections.

Let X be a metric space and T : X → X continuous. Consider the following two conditions: (TT) for every pair of nonempty open sets U and V in X, there is a positive

integer n such that T^{−n}(V) ∩ U ≠ ∅, (DO) there is a point x ∈ X such that the orbit of x is dense in X. As usual, we adopt the condition (TT) as the definition of topological transitivity, but note that some authors take (DO) instead. Any point with dense orbit is called a transitive point. A point which is not transitive is called intransitive. The set of transitive or intransitive points of (X, T) will be denoted by X_{tr} or X_{intr} respectively.

The two conditions are independent in general. Nevertheless, under some additional assumptions on the phase space (or on the map) the two definitions (TT)

Typeset by AMS-TEX

2000 Mathematics Subject Classification. Primary 37B05, 54H20, Secondary 37B20.

Key words and phrases. Topological transitivity, minimality, topological (weak) mixing, chaos, sensitivity, Li-Yorke pairs, topological entropy, specification property, proximal maps, spatiotemporal chaos.

and (DO) are equivalent. In fact, we have: if X has no isolated point then (DO) implies (TT); if X is separable and second category then (TT) implies (DO) [Silv].

Let us also remark that in general compact metric spaces, (TT) and (DO) are equivalent for onto maps. If T is transitive (i.e.,it satisfies (TT)) then T is onto. If a compact metric space admits a transitive map (i.e., if there exists a continuous selfmap T of X satisfying (TT)) then X has no isolated point if and only if it is infinite.

Convention: Throughout the paper, a couple (X, T) will denote a (topological) dynamical system. This means that X is an infinite compact metric space without isolated points and T : X → X is a surjective continuous map.

Let us note that a system (X, T) is transitive if and only if for every pair of non-empty open sets U and V in X the sequence of visit times n(U, V) := {n ≥ 0 : T^{−n}(V) ∩ U ≠ ∅} is infinite. If n ∈ ℤ_{+} and T^{n} is transitive then trivially T is transitive. The converse is not true in general. (X, T) is called totally transitive if the system (X, T^{n}) is transitive for all n ∈ ℤ_{+}.

The most fundamental dynamical systems are the minimal ones. These are systems which have no nontrivial subsystems. The fact discovered by G. D. Birkhoff is that any topological dynamical system (X, T) has minimal subsystems (M, T). The sets M such that (M, T) is minimal are called the minimal sets of (X, T). So, a subset M of X is minimal if it is closed, nonempty, invariant (i.e., T (M) ⊆ M) and contains no proper subset with these three properties. A nonempty closed set M ⊆ X is minimal if and only if the orbit of every point of M is dense in M. A point is called minimal or almost periodic if it belongs to a minimal set, and a dynamical system (X, T), or the map T, is called minimal if the set X is minimal. The set of all minimal points of (X, T) we define by X_{min}.

Given a dynamical system (X, T), the ω-limit set of a point x ∈ X under T, ω_{T} (x), is the set of all limit points of the trajectory of x, i.e., y ∈ ω_{T} (x) if and only if T^{n_{k}}(x) → y for some sequence of integers n_{k} ↗ ∞.

A point x ∈ X is called recurrent if for some sequence of integers n_{i} ↗ ∞, lim T^{n_{i}}x = x; clearly x ∈ ω_{T} (x) and every transitive point is recurrent. A dynamical system (X, T), or the map T, is called recurrent if any point x is recurrent; clearly any minimal dynamical system is also recurrent.

In a system (X, T) there are the following possibilities:

(a) X_{tr} = ∅, X_{intr} = X,

(b) X_{tr} is dense G_{δ} and (b1) X_{intr} = ∅ (minimality) or (b2) X_{intr} is dense.

and no other possibility exists ([Ki], see also [Ak], [AAB], [KS] and [DY]). In other words if (X, T) is a transitive system then the set of points that are not transitive is either empty or dense (equivalently: if the set of transitive points has nonempty interior then the system is minimal).

We will use some important families of transitive systems. Recall their definitions and some known facts. Let (X, T) be a transitive system. We say that the system (X, T) is

(1) an F-system if it is totally transitive and the set of periodic points is dense in X;

(2) a P-system if it is transitive and the set of periodic points is dense in X; (3) a ToP-system if it is non-minimal and each point of X is either transitive

or periodic;

(4) an M-system if it is non-minimal transitive and the set X_{min} is dense in X;

(5) a ToM-system if it is non-minimal and each point of X is either transitive or minimal;

(6) an E-system if it has a T -invariant probability measure on X, which is positive on every non-empty open set.

It is known that (1) ⟹ (2) ⟹ (4) ⟹ (6) (see [GW]). Of course, it is clear also that (3) ⟹ (2), (5) ⟹ (4) (see [DY]).

A dynamical system (X, T) is – an ergodic system if it has an ergodic T -invariant measure μ on X with full support (i.e., suppμ = X); – a syndetic system if for each nonempty open set U in X the sequence of return times n(U, U) is syndetic (i.e., n(U, U) has uniformly bounded gaps in ℤ_{+}).

An example in [W] shows that an E-system need not to be ergodic (in fact, the example is a P-system). Any minimal systems are ergodic systems and syndetic systems (see [GW], [GW2]).

Two topological dynamical systems (X, T) and (Y, S) are called disjoint if there exists no proper closed T × S–invariant subset of the cartesian product X × Y with projections X and Y on the two coordinates; such a subset is called a joining. Recall that when two systems are disjoint, at least one of them is minimal.

A pair {x, y} ⊆ X such that lim inf_{n→∞} ϱ(T^{n}x, T^{n}y) > 0 is said to be distal, and one for which lim_{n→∞} ϱ(T^{n}x, T^{n}y) = 0 is said to be asymptotic. A pair {x, y} is proximal if there exists a sequence n_{i} with lim_{i→∞} ϱ(T^{n_{i}}x, T^{n_{i}}y) = 0. A pair {x, y} ⊆ X is a Li–Yorke pair if simultaneously

lim inf_{n→∞} ϱ(T^{n}x, T^{n}y) = 0 and lim sup_{n→∞} ϱ(T^{n}x, T^{n}y) > 0.

Let us repeat these definitions for the case when X is a compact Hausdorff space. We denote by Δ the diagonal {(x, x) ∈ X^{2} : x ∈ X} of X^{2}. A pair {x, y} ⊆ X such that ω_{T ×T} (x, y) ∩ Δ = ∅ is distal, and one for which ω_{T ×T} (x, y) ⊆ Δ is asymptotic. A pair {x, y} is proximal if the set ω_{T ×T} (x, y) ∩ Δ is non-empty. A pair {x, y} ⊆ X is a Li–Yorke pair if the set ω_{T ×T} (x, y)∩Δ is non-empty and ω_{T ×T} (x, y) ⊈ Δ. Thus a pair {x, y} is a Li–Yorke pair if and only if it is proximal but not asymptotic. The sets of distal pairs, Li–Yorke pairs and asymptotic pairs partition X^{2}.

One of deep theorems of Topological Dynamics says: any point x of a topological dynamical system (X, T) is proximal to some minimal point y ([Au1], [El]). As we will see later, what we have asked for a system to be spatiotemporally chaotic: any point x is proximal and not asymptotic to points y which are arbitrarily close to x has a direct connection to the theorem. On the other hand, spatiotemporal chaotic maps are expected to have very little in common with distal maps.

Denote the set of proximal pairs by PR(X, T), asymptotic pairs by AS(X, T) and the set of points which are asymptotic to a point x by AS(x, T). Then PR(X, T) = ∩_{n=1}^{∞} (∪_{i=1}^{∞}(T × T)^{−i}(Δ_{n})) and AS(X, T) = ∩_{n=1}^{∞} (∪_{k=1}^{∞} (∩_{i=k}^{∞}(T × T)^{−i}(\overline{Δ}_{n}))), where (Δ_{n})_{n=1}^{∞} is a countable basis of neighborhoods of the diagonal Δ. Hence the set PR(X, T) is a G_{δ} subset of X^{2} and in any transitive system the sets AS(X, T) and AS(x, T) are sets of the first category in X^{2} and in X, respectively (see [HY2]).

Let (X, T) be a transitive system. We say that (X, T) is – a distal system if every non–diagonal pair {x, y} (i.e., x ≠ y) is distal; – an almost distal system if it is without Li-Yorke pairs; – a point distal system if there is a point which is distal with any other point; – a proximal system if it is without distal pairs.

The paper is organized as follows. In Section 1, after introducing general definitions and background, we survey some existing definitions of chaos, explaining what is known about them and about their relations.

In Section 2 we consider a very recent definition of spatiotemporal chaos which is based on Li–Yorke pairs and has some common features with sensitivity. We show that all topologically mixing systems (Theorem 2.1), weakly mixing minimal systems (Corollary 2.4), proximal systems (Proposition 2.5) and also some classes of recurrent systems (Theorems 2.6 and Proposition 2.8) are spatiotemporally chaotic.

Acknowledgements. The author thanks François Blanchard, Eli Glasner and Xiangdong Ye for fruitful discussions. His special thanks go to L’ubomír Snoha for fruitful discussions and his useful remarks on a previous version of the paper. This paper was written during the author’s visit to the Max-Planck-Institut für Mathematik at Bonn. The support and the kind hospitality of the institute are very much appreciated.

1. On Various Definitions of Chaos

1.1 Lyapunov Instability of Trajectories, Sensitivity

One family of definitions of chaos is based on the instability of trajectories. Given ε > 0, the map T is called Lyapunov ε–unstable at x ∈ X if for every neighborhood U of x, there is y ∈ U and n ≥ 0 with ϱ(T^{n}x, T^{n}y) > ε; T is called unstable at x (or the point x itself is called unstable) if there is ε > 0 such that T is Lyapunov ε-unstable at x.

If a dynamical system is pointwise unstable sometimes there exists no ε > 0 such that all points are ε–unstable. But if in addition the system is transitive then pointwise instability (in fact the instability of a point with dense orbit) implies uniform pointwise instability, i.e., the existence of such a universal ε > 0; this last property is also called sensitive dependence on initial conditions (sensitivity for short).

Auslander and Yorke call chaotic a system that is both sensitive and transitive [AY]. The Devaney’s definition of chaos is Auslander and Yorke’s definition with one extra assumption of the existence of a dense set of periodic points [D]. Recently this definition became rather popular because of the following theorem (see [Silv],

[BBCDS] and [GW]): Let X be an infinite metric space and f : X → X be continuous. If f is transitive and has dense periodic points then f has sensitive dependence on initial conditions. (Thus, one of the three conditions in the Devaney’s definition is redundant.)

A point x ∈ X is called an equicontinuity point (or (X, T) is equicontinuous at x) if for every ε > 0 there is a δ > 0 such that whenever y ∈ X satisfies ϱ(x, y) < δ then ϱ(T^{n}(x), T^{n}(y)) < ε for all n ≥ 0. Of course if X is compact and every x ∈ X is an equicontinuity point then (X, T) is equicontinuous. A topologically transitive system is called almost equicontinuous if it has at least one equicontinuity point. If a transitive system (X, T) has no equicontinuous points then it is sensitive. Moreover a minimal system is either equicontinuous or sensitive (see [AY], [AAB] and [GW] for details).

Now it is known also that any non–invertible transitive systems, or non-minimal E-systems (in particular, M-systems) are chaotic in the sense of Auslander and Yorke (see [GW], [AAB]).

1.2 Li-Yorke Pairs

Now consider definitions of chaos based on Li–Yorke pairs. The definition of Li– Yorke chaos is based on ideas in [LY]. Recall that a pair of points {x, y} ⊆ X is said to be a Li–Yorke pair (with modulus δ) if one has simultaneously

lim sup_{n→∞} ϱ(T^{n}x, T^{n}y) = δ > 0 and lim inf_{n→∞} ϱ(T^{n}x, T^{n}y) = 0.

A set S ⊆ X is called scrambled if any pair of distinct points {x, y} ⊆ S is a Li–Yorke pair. Finally, a system (X, T) is called chaotic in the sense of Li and Yorke (or shortly Li-Yorke chaotic) if X contains an uncountable scrambled set. A system (X, T) is called densely Li-Yorke chaotic if X contains a dense, uncountable scrambled set. The first motivation for studying this notion comes from the theory of interval transformations. For such maps the existence of a Li–Yorke pair implies the existence of an uncountable scrambled set [KuS].

Recently it was proved that the transitive systems with at least one periodic point (in particular, P-systems) are Li–Yorke chaotic [HY2].

A dynamical system (X, T) is called generically or densely chaotic if the set of Li–Yorke pairs is residual or dense in X × X, respectively. These notions were studied first on the interval. Say, in [Sn1] several conditions equivalent to generic chaos have been found. One of them shows that, in a sense, on the interval there is not a big difference between generic chaos and transitivity (see also [Mu]). Here only recall that topological transitivity on the interval implies generic chaos but the converse is not true, since there are generically chaotic maps arbitrarily close to a constant map. Dense chaos does not imply generic chaos (even on the interval). Nevertheless, in [Sn2] it is shown that in a class of interval maps which contains all piecewise monotone maps the notion of dense chaos and that of generic chaos coincide.

Recently it was shown that any transitive generically chaotic system (X, T) is Li–Yorke chaotic [HY2].

It is interesting to compare Li–Yorke and Auslander–Yorke chaoticity; neither implies the other.

Sturmian systems provide the simplest example of minimal almost distal systems which are sensitive [BGKM]. Let Y be the 1–torus and S_{α} be the rotation by the irrational number α ∈ Y. Code the orbits of (Y, S_{α}) according to the closed cover C = {[0, 1 − α], [1 − α, 1]} of Y. Call X the closed set of all bi–infinite C–names and let σ be the shift. All points of Y have only one C–name in X except 0 and all points of its orbit, which have two each; on the other hand there is a continuous map φ: X → Y sending an infinite name to the point of Y it was constructed from, and S_{α} ∘ φ = φ ∘ σ.

It is not hard to check that if x ≠ y ∈ Y, then the pair (x, y) is distal unless φ(x) = φ(y); in the last case (x, y) is asymptotic under the actions of σ and σ^{−1}: (X, σ) is a finite–to–one asymptotic extension of the isometry (Y, S_{α}).

On the other hand, there are scattering systems (see Subsection 1.4) which are Li-Yorke chaotic but not sensitive to initial conditions [AG]. Let us also remark that, since every non sensitive minimal map is equicontinuous, in the minimal case it is obvious that Li–Yorke chaos implies Auslander–Yorke chaos.

1.3 Mixing

A system (X, T) is said to be topologically mixing if for any pair of nonempty open sets U and V in X there is a k ∈ ℤ_{+} such that the sequence of visit times n(U, V) contains the sequence k, k + 1, k + 2, ....

Let us also recall the definition of so-called specification property which was introduced by R. Bowen. (X, T) satisfies the strong specification property or is said to be an SSP -system if the following holds: for any ε > 0 there exists an integer M(ε) such that for any k ≥ 2, for any k points x_{1}, x_{2}, …, x_{k} ∈ X, for any integers a_{1} ≤ b_{1} < a_{2} ≤ b_{2} < ⋯ < a_{k} ≤ b_{k} with a_{i} − b_{i−1} ≥ M(ε) for 2 ≤ i ≤ k, and for any integer p with p ≥ M(ε) + b_{k} − a_{1}, there exists a point x ∈ X with T^{p}(x) = x such that ϱ(T^{n}(x), T^{n}(x_{i})) ≤ ε for a_{i} ≤ n ≤ b_{i}, 1 ≤ i ≤ k. (X, T) satisfies the specification property or is said to be an SP-system if the above condition holds for k = 2. D. Ruelle defines also a weak specification property: the same as strong, except that x is not required to be periodic (hence no condition on p).

This seems to be a very strong condition, but it is satisfied by many examples. All systems with the specification property are topologically mixing, E-systems, F-systems and the topological entropy of T is positive (see [DGS] for details).

A system (X, T) is topologically weakly mixing if its cartesian square (X ×X, T × T) is transitive. The next two very useful properties characterize weak mixing systems: T is (topologically) weakly mixing if and only if given nonempty open sets U and V of X there is n ≥ 0 such that T^{−n}(U) ∩ U ≠ ∅ and T^{−n}(V) ∩ U ≠ ∅ [Pet]; (X, T) is weakly mixing if and only if for every pair of nonempty open sets U and V in X the sequence of visit times n(U, V) is thick (i.e., n(U, V) contains arbitrary long intervals) [Fu].

The implications between various mixing conditions implying transitivity may be summarized as follows:

Mixing ⟹ Weakly mixing ⟹ Totally transitive ⟹ Transitive.

All of the implications in the diagram are elementary to check except the fact that weakly mixing maps are totally transitive (a proof can be found in [B]). Although the existence of plenty of maps which are weakly mixing but not mixing is well known (see for instance [GM], [AAB], [JS], [Cr]), sometimes they are difficult to exhibit for some classes of dynamical systems. Let us also remark that an F-system is weakly mixing ([Fu], [B]).

A subset E of X is called independent for (X, T) if for any finite collection x_{1}, x_{2}, …, x_{n} of distinct points of E the point (x_{1}, x_{2}, …, x_{n}) is transitive in the product system (X, T)^{n} ((X × ... × X, T × ... × T), n times). Obviously any independent set is also scrambled.

(Weak) mixing is a rather strong chaotic property. In every weakly mixing system there exists an uncountable independent set ([Iw], using the main theorem in [M]). It implies generic (hence Li–Yorke), and Auslander–Yorke chaos (see [BGKM]). None of the converse implications is true. Sturmian systems are Auslander–Yorke chaotic without being weakly mixing. In [Sn1] there are examples of interval maps showing that weak mixing is not implied by generic chaos either, even assuming transitivity. Another example can be found in [BGKM]: Let (X, T) be the full shift on 2 letters and (Y, S) be an irrational rotation of the unit circle; let (Z, R) be the factor of (X ×Y, T ×S) obtained by collapsing all elements of the form (x_{0}, y) where x_{0} is the fixed point on 0. One can check that (Z, R) is transitive but not weakly mixing and has a dense set of Li–Yorke pairs with modulus δ > 0, which is G_{δ}; on the other hand since it contains a fixed point it is Li–Yorke chaotic by [HY2].

1.4 Topological Chaos and Scattering

Positive topological entropy and scattering can be considered as a next family of chaotic properties, based on the complexity of covers.

The topological entropy of a map T ∈ C(X), denoted by h(T), was introduced in [AKM] as an invariant of topological conjugacy and an analogue to the notion of metric (measure-theoretical) entropy (see, e.g., [Wal]). Consider a topological dynamical system (X, T). For any finite cover C of the compact space X, let N (C) be the minimal cardinality of a sub-cover of C. The topological complexity function of the finite cover C of (X, T) is the non-decreasing function

c_{T} (C, n) = N (C^{n}),

where C^{n} = C ⋁ T^{−1}C ⋁ ... ⋁ T^{−(n−1)}C. The exponential growth rate of this function is the topological entropy of C, and the topological entropy of (X, T), h(X, T), is the supremum of the topological entropies of finite open covers (see [AKM]).

We give also here one of the definitions which were introduced by R. Bowen and E.I. Dinaburg (see, e.g., [Wal]) and are equivalent to the original one.

So let, as usual, (X, ϱ) be a compact metric space and let T be a continuous map from X into itself. A subset E of X is called (n, ε)-separated if for every two different points x, y ∈ E there exists 0 ≤ j < n with ϱ(T^{j}(x), T^{j}(y)) > ε. Let s_{n}(T, ε) be the maximal possible cardinality of an (n, ε)-separated set. Then the (topological) entropy of T is defined by

h(T) = lim_{ε→0} lim sup_{n→∞} \frac{1}{n} log s_{n}(T, ε) .

We see that the topological entropy represents the exponential growth rate for the number of orbit segments distinguishable with arbitrarily fine but finite precision. A dynamical system (X, T) is called topologically chaotic if its topological entropy is positive. A dynamical system (X, T) is called scattering if any finite cover C by non–dense open sets has unbounded complexity, i.e., c_{T} (C, n) → ∞, 2–scattering if the same condition holds but for 2–set covers only.

A system is scattering if and only if its cartesian product with any minimal system is transitive [BHM]. Weak mixing implies scattering; scattering is strictly weaker than weak mixing [AG], but the two properties are equivalent in the minimal case [BHM].

Scattering, actually even 2–scattering, implies transitivity [BHM], Li–Yorke chaos and generic chaos [HY2].

Scattering does not imply that there exist no equicontinuity points [AG]. The same example shows that scattering systems are not necessarily sensitive to initial conditions: a transitive sensitive system has no equicontinuity points [AAB].

If (X, T) is transitive but not sensitive, T has a property called uniform rigidity ([GW], [AAB]). It implies that T is a homeomorphism and that it has zero entropy. Therefore any topologically chaotic transitive system has sensitive dependence on initial conditions; it is thus Auslander–Yorke chaotic.

Recall that a minimal scattering system is weakly mixing, therefore chaotic in many senses. There are however many examples of weakly mixing systems with zero entropy (for example, any uniquely ergodic system which is, measure theoretically, weakly mixing and has simple spectrum) and even minimal and uniformly rigid ones [GM]. Topological chaos is another strong notion of chaoticity.

Very recently a positive answer to the question whether topological chaos implies Li–Yorke chaos was given in [BGKM]. It is worth mentioning that Li–Yorke pairs are not the only ones that are necessarily found in topological chaotic systems: there must also exist asymptotic pairs, that is, pairs (x, y) such that d(T^{n}x, T^{n}y) → 0 as n → ∞ [BHR]. By the way let us also remark that there are topologically chaotic (transitive) systems without distal pairs.

In [BGKM] the authors have also explored systems without Li–Yorke pairs. Surely distal systems have this property but, as we shall see, the class of such systems is much wider. However, it turns out that systems without Li–Yorke pairs share many of the basic properties of distal systems. We have already seen that they also have zero entropy; such systems are minimal when transitive, and any factor of a transitive system without Li–Yorke pairs has no Li–Yorke pairs (see [BGKM]). For these reasons the transitive systems without Li–Yorke pairs are called almost distal. One important difference with distality is that almost distal systems are not necessarily invertible. The Sturmian and Morse subshifts, when considered only on one–sided sequences, are non–invertible almost distal systems (see [BGKM]). Wide classes of almost distal systems which are almost one–to–one, not finite–to–one, asymptotic extensions of rotations on the 2-torus which are minimal point distal homeomorphisms (respectively minimal point distal noninvertible maps) on that torus are described in [R] (respectively [KST]).

As a generalisation of results in [Fu] and [BHM], in [BGKM] it was shown that almost distal minimal systems are also disjoint from any scattering system.

1.5 Comparing of the Definitions of Chaos

Recall that for interval maps many of the existing definitions of chaos more or less coincide (see for instance [KS]).

Below we summarize the relations between some definitions of chaos for transitive dynamical systems (no other implications work except of those which can be deduced using the transitivity of implication). We also add short remarks, inspired by the results mentioned above.

\xymatrix{Mixing \ar@{=>}[r] & Weakly mixing \ar@{=>}[r] \ar@{=>}[d] & Scattering \ar@{=>}[r] & Generic chaos \ar@{=>}[d] \\  & Sensitivity & Topological chaos \ar@{=>}[l] \ar@{=>}[r] & Li–Yorke chaos}

If a dynamical system is minimal, then it is known that some of the definitions of chaos are equivalent. Let us remark some very useful properties of minimal weakly mixing systems. Firstly, any minimal system which is not weakly mixing has a non-trivial equicontinuous factor (see [Au] for a simple proof due to McMahon). In [BHM] it was shown by using this property that for a minimal dynamical system scattering and weak mixing are equivalent. Secondly, for minimal systems the notion of weak mixing is equivalent to the statement that RP = X × X, where RP is the regionally proximal relation on X (the regionally proximal relation RP (X, T) is the set of all points (x, y) of X^{2} such that for each ε > 0 and each open neighborhoods U of x and V of y there are x′ ∈ U, y′ ∈ V and n ≥ 0 with ϱ(T^{n}(x′), T^{n}(y′)) < ε) (see [Pet]). Recently in [HY2] it was shown more. Namely that for minimal systems the notions of weak mixing, generic chaos and dense Li-Yorke chaos are equivalent. In the below diagram we summarize the relations between the chaotic systems described in the previous diagram (moreover, we add dense Li–Yorke chaos).

[figure: Mixing; Weakly mixing; Scattering; Generic chaos; Sensitivity; Li–Yorke chaos; Dense Li–Yorke chaos; Topological chaos]

In the below table, we indicate some transitive dynamical systems and their belonging to (denoted as ⊊), or non-empty intersection with (denoted as ∩) the sets of chaotic systems which was described in the above diagrams: 1 - topologically chaotic; 2 - topologically mixing; 3 - weakly mixing; 4 - scattering; 5 - generically chaotic; 6 - Li-Yorke chaotic; 7 - systems with sensitive dependence on initial conditions. As usual the symbol ? means that the author does not know anything about the connection.

1 2 3  4 5 6  7
SP-systems  ⊊ ⊊ ⊊  ⊊ ⊊ ⊊  ⊊
F-systems  ∩ ⊊ ⊊  ⊊ ⊊ ⊊  ⊊
P-systems  ∩ ∩ ∩  ∩ ∩ ⊊  ⊊
M-systems  ∩ ∩ ∩  ∩ ∩ ?  ⊊
ToP-systems  ∩ ? ?  ? ? ⊊  ⊊
ToM-systems  ∩ ? ?  ? ? ?  ⊊

2. ST Chaos

Other definitions, introduced in [BGKM], based on Li–Yorke pairs but having some common features with sensitivity may also be considering. We say that a dynamical system (X, T) has chaotic dependence on initial conditions if there is δ > 0 such that for any x ∈ X and every neighborhood U of x there is y ∈ U such that the pair (x, y) is Li–Yorke with modulus δ.

A dynamical system (X, T) is said to be spatiotemporally strongly chaotic or ST strongly chaotic for short if it is transitive and depends chaotically on initial conditions.

ST strong chaos implies Auslander–Yorke chaos by definitions. It seems that the assumption of the existence of a universal δ > 0 in the definition of ST strong chaos is a too strong one. So, we will say that a dynamical system (X, T) is spatiotemporally chaotic or ST chaotic for short if it is transitive and for any x ∈ X and every neighborhood U of x there is y ∈ U such that the pair (x, y) is Li–Yorke.

What are the connections between ST chaos and other definitions of chaos? What topological properties of a system imply ST chaos or are implied by the ST chaos?

Although topologically chaotic transitive systems are Li–Yorke chaotic and Auslander–Yorke chaotic, they do not imply ST chaos: there are point distal maps (which cannot be ST strong chaotic) with positive topological entropy.

If (X, T) is a dynamical system and S is an increasing sequence of positive integers, denote by Prox(x, T) the set of all points proximal to x and by Prox^{S}(x, T) the set of all points which are proximal to x with respect to the sequence S (i.e., y ∈ Prox^{S}(x, T) if and only if there is a subsequence n_{1} < n_{2} < … of the sequence S such that T^{n_{i}}(x) converges to x).

Lemma 2.0. Let (X, T) be a dynamical system and x ∈ X. If Prox(x, T) is dense then there is an increasing sequence S of positive integers such that Prox^{S}(x, T) is G_{δ}-dense. Hence the set Prox(x, T) is residual.

Proof. We have a dense set {p_{λ} : λ ∈ Λ} (Λ being an index set) such that each of the points p_{λ} is proximal to x. Thus, for every λ there si an increasing sequence S_{λ} of positive integers such that the trajectory of p_{λ} relative to this sequence converges to x. Now let n_{1} < n_{2} < … be the increasing sequence of positive integers containing all elements of all sequences S_{λ}, λ ∈ Λ. Denote this sequence by S. Obviously, all the points p_{λ} are proximal to x also with respect to the sequence S. Thus, the set Prox^{S}(x, T) is dense. To finish the proof we need to show that it is of type G_{δ}. This follows from the obvious fact that Prox^{S}(x, T) = ∩_{k=1}^{∞}(∪_{j=k}^{∞}T^{−n_{j}}(W_{k})) where (W_{k})_{k=1}^{∞} is a countable basis of neighborhoods of x.

Theorem 2.1. Any topologically mixing system (X, T) is spatiotemporally chaotic.

Proof. ^{1} Fix a point x ∈ X. Since X is compact, there are a point x_{∞} ∈ ω_{T} (x) and a sequence n_{i} ↗ ∞ such that lim_{i→∞} T^{n_{i}}(x) = x_{∞}. Denote the sequence {n_{i}}_{i=1}^{∞} by N_{x}^{x_{∞}} and for k = 1, 2, … let V_{k} be the open ball with radius 1/k centered at x_{∞}. Let U be a non-empty open set in X. Since T is mixing, there is n_{i_{1}} ∈ N_{x}^{x_{∞}} such that T^{−n_{i_{1}}} (V_{1}) ∩ U ≠ ∅ and therefore there is a non-empty open set U_{1} ⊂ U such that T^{n_{i_{1}}} (U_{1}) ⊂ V_{1}. Similarly it is possible to find n_{i_{2}} ∈ N_{x}^{x_{∞}}, i_{2} > i_{1}, such that T^{−n_{i_{2}}} (V_{2}) ∩ U_{1} ≠ ∅ and therefore there is a non-empty open set U_{2} ⊂ U_{1} such that T^{n_{i_{2}}} (U_{2}) ⊂ V_{2}. By induction we obtain a subsequence {n_{i_{k}}}_{k=1}^{∞} of N_{x}^{x_{∞}} and a nested sequence of non-empty open sets {U_{k}}_{k=1}^{∞} in U such that T^{n_{i_{k}}} (U_{k}) ⊂ V_{k} for every k. Hence the set ∩_{k=1}^{∞}\overline{U}_{k} is non-empty and for every y in this set we have lim_{k→∞} T^{n_{i_{k}}} (y) = x_{∞}. In other words the pair {x, y} is proximal.

Since U was an arbitrary open set, the set Prox(x, T) is dense in X. By Lemma 2.0 this set is also residual. Since, as we mentioned in Section 0, the set AS(x, T) (of points which are asymptotical to the point x) is the first category, the set of points y such that {x, y} is a Li–Yorke pair is residual. In particular, such points exist in any neighborhood of x.

1One can prove this theorem also by using a result from [JS] (see Concluding Remarks).

Corollary 2.2. Any system which has the specification property is spatiotemporally chaotic.

Now a natural question is whether even all weakly mixing systems are spatiotemporally chaotic. We do not know the answer. Nevertheless, we are going to show that all weakly mixing minimal systems are necessarily spatiotemporally chaotic.

Recall that if (X, T) is weakly mixing, for every pair of nonempty open sets U and V in X the sequence of visit times n(U, V) is thick (i.e., n(U, V) contains arbitrary long intervals). A point x is minimal if it belongs to a minimal set of X. If x is minimal, for any its neighborhood U the sequence of return times n(x, U) := {n ≥ 0 : T^{n}(x) ∈ U} is syndetic (i.e., n(x, U) has uniformly bounded gaps in ℤ_{+}).

Lemma 2.3. Let (X, T) be a weakly mixing system. Then for any minimal point x the set of points which form Li–Yorke pairs with x is dense in X.

Proof. By G. D. Birkhoff any topological dynamical system (X, T) has a minimal set. Fix a minimal point x ∈ M, where M is a minimal subset of X, and a nonempty open set U. We are going to find a point y ∈ U which is proximal to x.

Let {V_{k}}_{k=1}^{∞} be the sequence of open balls with radius 1/k centered at x. Since T is weakly mixing and x is minimal, there is n_{1} such that T^{−n_{1}}(V_{1}) ∩ U ≠ ∅ and T^{n_{1}}(x) ∈ V_{1}. Hence there is a non-empty open set U_{1} ⊂ U such that T^{n_{1}}(U_{1}) ⊂ V_{1}. Again thus it is possible to find n_{2} > n_{1} such that T^{−n_{2}}(V_{2}) ∩ U_{1} ≠ ∅ and T^{n_{2}}(x) ∈ V_{2}. Therefore there is a non-empty open set U_{2} ⊂ U_{1} such that T^{n_{2}}(U_{2}) ⊂ V_{2}. By induction we obtain an increasing sequence {n_{k}}_{k=1}^{∞} of positive integers and a nested sequence of non-empty open sets {U_{k}}_{k=1}^{∞} in U such that T^{n_{k}}(U_{k}) ⊂ V_{k} for every k and lim_{k→∞} T^{n_{k}}(x) = x. Hence the set ∩_{k=1}^{∞}\overline{U}_{k} is non-empty and for every y in this set we have lim_{k→∞} T^{n_{k}}(y) = x. In other words, the pair {x, y} is proximal.

We showed that the set Prox(x, T) of all points which are proximal to x is dense in X. By Lemma 2.0 this set is residual. Since the set AS(x, T) is the first category, the set of points y such that {x, y} is a Li–Yorke pair is residual. In particular, such points exist in any neighborhood of x.

Corollary 2.4. Any weakly mixing minimal system (X, T) is spatiotemporally chaotic.

Since for minimal systems the notions of weak mixing, scattering, generic chaos and dense Li-Yorke chaos are equivalent, minimal systems with any of these properties are also ST chaotic.

Recal that a transitive system is proximal if there are no distal pairs.

Proposition 2.5. Any proximal transitive system is spatiotemporally chaotic.

Proof. It is a direct corollary of the definition of a proximal system and the fact that for any transitive system (X, T) and any point x ∈ X the set AS(x, T) of points which are asymptotical to the point x is of the first category.

A system (X, T) is called uniformly rigid if there is a sequence n_{i} ↗ ∞ such that the sequence {T^{n_{i}}} tends uniformly to the identity map on X. In particular, it means that any point of X is recurrent. One can check that topological mixing systems are not uniformly rigid (see also [GM]). Although any uniformly rigid systems are zero topological entropy and without asymptotic pairs, some of them are weakly mixing systems (see [GW], [AAB]). We would like to note the following implications for transitive systems

(X, T) is topologically chaotic ⟹ AS(X, T) is non-empty ⟹ (X, T) is not uniformly rigid ⟹ (X, T) is Auslander–Yorke chaotic

([BHR], [GM], [AAB]).

From [AAB] we know that there is a uniformly rigid system (X, T) with a fixed point u ∈ X which is a unique minimal subset of X (the system is a subsystem of the so called Bebutov system (see also [N], [KW])).

Theorem 2.6. Let (X, T) be a uniformly rigid transitive system. Suppose there is a fixed point u which is a unique minimal subset of X. Then (X, T) is spatiotemporally chaotic.

Proof. Since the system is transitive by the assumption, we need to show only that for any x ∈ X and every neighborhood U of x there is y ∈ U such that the pair (x, y) is Li–Yorke.

Since (X, T) is uniformly rigid there is a sequence n_{i} ↗ ∞ such that for any x the sequence {T^{n_{i}}(x)} tends to x. Hence each point (x, y) ∈ X^{2} is recurrent for the map T ×T. Since the ω-limit set of x is a non-empty closed ivariant set, by Birkhoff theorem there is a minimal subset in it and by the assumpion of the theorem it is the fixed point u.

Fix a point x ∈ X. If x coincides with the fixed point u then it is obvious that any pair {x, y}, where x ≠ y, is Li–Yorke. Suppose x ≠ u and let U be a neighborhood of x. Since x is a recurrent point there exists a positive integer n such that T^{n}(x) ∈ U. Since u ∈ ω_{T} (x) and T is continuous, there exists a sequence n_{i} ↗ ∞ such that lim_{i→∞}(T^{n_{i}}(x), T^{n+n_{i}}(x)) = (u, u). Therefore the pair {x, T^{n}(x)} as a recurrent point of T × T is also Li–Yorke.

Repeat that any topologically transitive system which is not sensitive is uniformly rigid and almost equicontinuous. An example of uniformly rigid systems in [AAB] is an almost equicontinuous system with a fixed point which is a unique minimal subset of X. Therefore there are spatiotemporally chaotic systems which are not sensitive.

Any minimal system is either equicontinuous or sensitive (see [AAB]). As a corollary of this fact we have

Proposition 2.7. Any minimal spatiotemporally chaotic system is sensitive.

Recall that a set is called regular closed or canonical closed if it is the closure of an open set (equivalently, if it is the closure of its interior). Suppose we have a collection X := {X_{0}, X_{1}, ..., X_{n−1}} of subsets of X such that T (X_{i}) ⊆ X_{i+1(mod n)} for 0 ≠ i < n − 1. Clearly, T^{k}(X_{i}) ⊆ X_{i+k(mod n)} for all k ≥ 0. A collection X is called regular periodic decomposition if the X_{i} are regular closed, X_{i} ∩ X_{j} is nowhere dense whenever i ≠ j and ∪_{i=0}^{n−1}X_{i} = X. From [B] we know that a map T is transitive if and only if T^{n} is transitive on each element of a regular periodic decomposition of T of length n and any transitive map T has at most one regular periodic decomposition of length n.

From [DY] we know that there are transitive non-minimal systems any nontransitive point of which is periodic, i.e. ToP-systems.

Proposition 2.8. Any ToP-system is spatiotemporally chaotic.

Proof. Let (X, T) be a ToP-system. Fix a point x ∈ X and its neighborhood U. Suppose x is a periodic point of period n. Then by [B] there is a regular periodic decomposition {X_{0}, X_{1}, ..., X_{n−1}} of period n. Since x ∈ X_{i} for some 0 ≤ i < n−1, the map G := T^{n} : X_{i} → X_{i} is transitive and x is a fixed point of G. Since X_{i} is a regular closed set, the set U ∩ X_{i} has non-empty interior. Hence a pair {x, y}, where y ∈ U ∩ X_{i} is a transitive point of G (consequently of T), is Li–Yorke.

Let x be a transitive point of (X, T) and U be its neighborhood. Since T is a ToP-system, there is a periodic point y of some period n. Then by [B] there is a regular periodic decomposition {X_{0}, X_{1}, ..., X_{n−1}} of period n such that x ∈ X_{i} and x is transitive for the map G := T^{n} : X_{i} → X_{i}. Since the transitive map T has at most one regular periodic decomposition of length n, there is a periodic point y_{0} ∈ X_{i} of the period n (from the T -orbit of the periodic point y).

As we know X_{i} is a regular closed set. Hence int(U ∩ X_{i}) is non-empty and by transitivity of G there is a positive integer m such that G^{m}(x) ∈ U ∩ X_{i}. Since y_{0} ∈ ω_{G}(x) and G is continuous, there exist a sequence m_{i} ↗ ∞ such that lim_{i→∞}(G^{m_{i}}(x), G^{m+m_{i}}(x)) = (y_{0}, y_{0}). The pair {x, G^{m}(x)} is also recurrent point of G × G, because the point x is transitive. Hence {x, G^{m}(x)} is Li–Yorke for the map T.

Concluding Remarks. Among other new results in Topological Dynamics we wish to mention two directions which have common features with the subject of the article.

(i) - Stroboscopical property. If S is an increasing sequence of positive integers {n_{k}}_{k=1}^{∞}, denote the set of all limit points of the sequence T^{n_{1}}(x), T^{n_{2}}(x), T^{n_{3}}(x), … by ω_{T} (x, S) (in particular, if S = ℕ it is the ω-limit set of x with respect to T (i.e., ω_{T} (x) = ω_{T} (x, ℕ))). Note that since X is compact, ω_{T} (x, S) is nonempty. A system (X; T) has the (Misiurewicz) strong stroboscopical property if, given an increasing sequence of positive integers S and a point x ∈ X, one can always find a dense set of points y ∈ X with x ∈ ω_{T} (y, S). This and similar notions have very recently been studied in [JS]. Among many other results, it was shown that the notions of topological mixing and strong stroboscopical property are equivalent. We proved Theorem 2.1 by using only the definition of topological mixing. Nevertheless, the following proof which uses this results from [JS] is much shorter. Fix x ∈ X and take a sequence n_{i} ↗ ∞ such that T^{n_{i}}(x) converges to a point x_{1} ∈ X. Fix a point x_{2} ∈ X, x_{2} ≠ x_{1} (we may assume that ϱ(x_{2}, x_{1}) ≥ 1/2 diam(X)). By [JS; Proposition 5], for j = 1, 2 there is a G_{δ}-dense set Y_{j} of points whose ω-limit set with respect to the sequence {n_{i}}_{i=1}^{∞} contains the point x_{j}. Put Y = Y_{1} ∩ Y_{2}. For any y ∈ Y, {x, y} is obviously a Li-Yorke pair with modulus δ = ϱ(x_{1}, x_{2}) (thus, we may claim that δ ≥ 1/2 diamX). Since Y is G_{δ}-dense, the result follows. We proved even more - any topologically mixing system (X, T) is spatiotemporally strong chaotic. (ii) - The sequence of visit/return times. As we mentioned, the definitions of transitive, weakly mixing and mixing systems are equivalent with saying that the sequences of visit times n(U, V) := {n ≥ 0 : T^{−n}(V) ∩ U ≠ ∅} are infinite, thick or contain the sequence k, k + 1, k + 2, … for some positive integer k, respectively. It was very recently shown that for any E-system the sequence of visit times is syndetic [GW2], for any scattering system and strong scattering system (systems which are disjoint from any E-system) the sequence of visit times is a recurrent set and a Poincaré set, respectively (see [HY3] for details).

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Institute of Mathematics, NASU, Tereshchenkivs’ka 3, 01601 Kiev, Ukraine

   E-mail address: skolyada@imath.kiev.ua