Dynamical compactness and sensitivity

To link the Auslander point dynamics property with topological transitivity, in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system $(X,T)$ given by a compact metric space $X$ and a continuous surjective self-map $T:X \to X$. Observe that each weakly mixing system is transitive compact, and we show that any transitive compact M-system is weakly mixing. Then we discuss the relationships among it and other several stronger forms of sensitivity. We prove that any transitive compact system is Li-Yorke sensitive and furthermore multi-sensitive if it is not proximal, and that any multi-sensitive system has positive topological sequence entropy. Moreover, we show that multi-sensitivity is equivalent to both thick sensitivity and thickly syndetic sensitivity for M-systems. We also give a quantitative analysis for multi-sensitivity of a dynamical system.

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title: Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation
abstract: 0.  The objects of ergodic theory -measure spaces with measure-preserving transformation groups-wil l be called processes, those of topological dynamics-compact metric spaces with groups of homeomorphisms-will be called flows. We shall be concerned with what may be termed the "arithmetic" of these classes of objects. One may form products of processes and of flows, and one may also speak of factor processes and factor flows. By analogy with the integers, we may say that two processes are relatively prime if they have no non-trivial factors in common. An alternative condition is that whenever the two processes appear as factors of a third process, then their product too appears as a factor. In our theories it is unknown whether these two conditions are equivalent. We choose the second of these conditions as the more useful and refer to it as disjointness. Our first applications of the concept of disjointness are to the classification of processes and flows. It will appear that certain classes of processes (flows) may be characterized by the property of being disjoint from the members of other classes of processes (flows). For example the processes with entropy 0 are just those which are disjoint from all Bernoulli flows. Another application of disjointness of processes is to the following filtering problem. If {xn} and {Yn} represent two stationary stochastic processes, when can {xn} be filtered perfectly from {Xn + Yn}? We will find (Part I, §9) that a sufficient condition is the disjointness of the processes in question. For flows the principal application of disjointness is to the ~tudy of properties of minimal sets (Part III). Consider the flow on the unit circle K = {z: [zl = 1 } that arises from the transformation z --~ z 2. What can be said about the "size" of the minimal sets for this flow, that is, closed subsets of K invariant under z ~ z ~, but not containing proper subsets with these properties. Uncountably many such minimal sets exist in K. Writing z = exp (2~ri Ean/2n), an = 0, 1, we see that this amounts to studying the mini-

title: Stroboscopical property in topological dynamics
abstract: A dynamical system given by a metric space and its continuous selfmap is said to have the (Misiurewicz) stroboscopical property if, given any point z from the space and any increasing sequence of positive integers, there is a point whose ω -limit set relative to this sequence contains  z . In the paper it is shown that some minimal skew product homeomorphisms on the torus do not have such property. Nevertheless, we show that some important classes of systems do have the stroboscopical property. On the other hand, there is a Devaney chaotic and topologically weakly mixing subshift without the stroboscopical property.

title: Li-Yorke sensitivity
abstract: We introduce and study a concept which links the Li–Yorke versions of chaos with the notion of sensitivity to initial conditions. We say that a dynamical system (X,T) is Li–Yorke sensitive if there exists a positive e such that every xX is a limit of points yX such that the pair (x,y) is proximal but not e-asymptotic, i.e. for infinitely many positive integers i the distance ρ(Ti(x),Ti(y)) is greater than e but for any positive δ this distance is less than δ for infinitely many i.

title: Classifying dynamical systems by their recurrence properties
abstract: In his seminal paper of 1967 on disjointness in topological
dynamics and ergodic theory H. Furstenberg started a systematic
study of transitive dynamical systems. In recent years this work
served as a basis for a broad classification of
dynamical systems by their recurrence properties. In this paper 
I describe some aspects of this new theory and its connections
with combinatorics, harmonic analysis and the theory of
topological groups.

title: Mixing and proximal cells along sequences
abstract: A dynamical system (X, T) is -transitive if for each pair of open and non-empty subsets U and V of X, , where is a collection of subsets of that is hereditary upward. (X, T) is -mixing if (X × X, T × T) is -transitive. For a subset S of , (x, y) X × X is S-proximal if and the S-proximal cell PS(x) is the set of points that are S-proximal to x X. We show that if (X, T) is -mixing, then for each (the dual family of ) and x X, PS(x) is a dense Gδ subset of X, and when (X, T) is minimal and is a filter the reciprocal is true. Moreover, other conditions under which the reciprocal is true are obtained. Finally the structure of proximal cells for -mixing systems is discussed, and a new and simpler proof of the Xiong–Yang theorem is presented.

title: F-mixing and weak disjointness
abstract: A topological system (X, f ) is F-transitive if for each pair of opene subsets U and V of X, Nf (U, V ) ={ n ∈ Z+: f n U ∩ V � } ∈F ,w hereF is a collection of subsets of Z+ which is hereditary upward. (X, f ) is F-mixing if (X × X, f × f) is F-transitive. In this paper F-mixing systems are characterized in terms of the chaoticity of the systems. Moreover, weak disjointness is studied via family. We will give conditions such that a dual theorem of the Weiss–Akin–Glasner theorem holds. Examples with this dual theorem fails for some “good” families are obtained.

title: Proximity and distality via Furstenberg families
abstract: In this paper proximity, distality and recurrence are studied via Furstenberg families. A new proof of some classical results on the conditions when a proximal relation is an equivalence one is given. Moreover, for a family F , F -almost distality and F -semi-distality are defined and characterized. As an application a new characterization of PI-flows is obtained.

title: Transitive points via Furstenberg family
abstract: Let ( X , T ) be a topological dynamical system and F be a Furstenberg family (a collection of subsets of Z + with hereditary upward property). A point x ∈ X is called an F -transitive one if { n ∈ Z + : T n x ∈ U } ∈ F for every non-empty open subset U of X ; the system ( X , T ) is called F -point transitive if there exists some F -transitive point. In this paper, we aim to classify transitive systems by F -point transitivity. Among other things, it is shown that ( X , T ) is a weakly mixing E-system (resp. weakly mixing M-system, HY-system) if and only if it is {D-sets}-point transitive (resp. {central sets}-point transitive, {weakly thick sets}-point transitive). It is shown that every weakly mixing system is F i p -point transitive, while we construct an F i p -point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is Δ ⁎ ( F w t ) -transitive if and only if it is weakly disjoint from every P-system.

title: Recent development of chaos theory in topological dynamics
abstract: We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li–Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.

title: Minimal flows and their extensions
abstract: Flows and Minimal Sets. Equicontinuous Flows. The Enveloping Semigroup of a Transformation Group, I. Joint Continuity Theorems. Distal Flows. The Enveloping Semigroup, II. The Furstenberg Structure Theorem for Distal Minimal Flows. Universal Minimal Flows and Ambits. The Equicontinuous Structure Relation and Weakly Mixing Flows. The Algebraic Theory of Minimal Flows. Disjointness. Invariant Measures on Flows. Kakutani-Bebutov Theorems. General Structure Theorems.

title: Sensitive dependence on initial conditions
abstract: It is shown that the property of sensitive dependence on initial conditions in the sense of Guckenheimer follows from the other two more technical parts of one of the most common recent definitions of chaotic systems. It follows that this definition applies to a broad range of dynamical systems, many of which should not be considered chaotic. We investigate the implications of sensitive dependence on initial conditions and its relation to dynamical properties such as rigidity, ergodicity, minimality and positive topological entropy. In light of these investigations and several examples which we exhibit, we propose a natural family of dynamical systems- chi -systems-as a better abstract framework for a general theory of chaotic dynamics.

title: Residual properties and almost equicontinuity
abstract: A propertyP of a compact dynamical system (X,f) is called a residual property if it is inherited by factors, almost one-to-one lifts and surjective inverse limits. Many transitivity properties are residual. Weak disjointness from all propertyP systems is a residual property providedP is a residual property stronger than transitivity. Here two systems are weakly disjoint when their product is transitive. Our main result says that for an almost equicontinuous system (X, f) with associated monothetic group Λ, (X, f) is weakly disjoint from allP systems iff the onlyP systems upon which Λ acts are trivial. We use this to prove that every monothetic group has an action which is weak mixing and topologically ergodic.

title: On Lyapunov exponent and sensitivity
abstract: Sensitive dependence on initial conditions is widely understood as being the central idea of chaos. For a large class of transformations of the interval, we prove that positiveness of the Lyapunov exponent implies the sensitivity property. We also provide bounds for the sensitivity constant.

title: Measure-theoretical sensitivity and equicontinuity
abstract: For an invariant measure µ in a topological dynamics, notions of µ-sensitivity, µ-complexity and µ-equicontinuity are introduced and investigated. It turns out that µ-sensitivity defined here is equivalent to pairwise sensitivity defined by Cadre and Jacob. For an ergodic µ, µ-equicontinuity, no µ-complexity pair and non-µ-sensitivity are equivalent, which implies minimality and equicontinuity when restricted to the support. Moreover, the notion of µ-sensitive set is introduced, it is shown that a transitive system with an ergodic measure of full support has zero topological entropy if there is no uncountable µ-sensitive set, and a non-minimal transitive system with dense minimal points has infinite sequence entropy for some sequence.For a minimal system (X, T) it is shown that (x1, x2) is regionally proximal if and only if, for any neighborhood U of x2, {n∈ℤ+:Tnx1∊U} is a Poincaré sequence. This implies that if (xi, xi+1) is regionally proximal for i = 1, …, n − 1, then (x1, …, xn) is n-regionally proximal. The structure of a minimal system is determined via the cardinalities of sensitive sets for µ.

title: Sensitive dependence to initial conditions for one dimensional maps
abstract: This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The equivalence classes of maps which display sensitivity to initial conditions for large sets of initial conditions are characterized.

title: Stronger forms of sensitivity for dynamical systems
abstract: For continuous self-maps of compact metric spaces, we initiate a preliminary study of stronger forms of sensitivity formulated in terms of 'large' subsets of . Mainly we consider 'syndetic sensitivity' and 'cofinite sensitivity'. We establish the following. (i) Any syndetically transitive, non-minimal map is syndetically sensitive (this improves the result that sensitivity is redundant in Devaney's definition of chaos). (ii) Any sensitive map of [0,1] is cofinitely sensitive. (iii) Any sensitive subshift of finite type is cofinitely sensitive. (iv) Any syndetically transitive, infinite subshift is syndetically sensitive. (v) No Sturmian subshift is cofinitely sensitive. (vi) We construct a transitive, sensitive map which is not syndetically sensitive.

title: Analogues of Auslander–Yorke theorems for multi-sensitivity
abstract: We study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. Our main result states that a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. This is an analog of the Auslander–Yorke dichotomy theorem: a minimal system is either sensitive or equicontinuous. Furthermore, we introduce the concept of a syndetically equicontinuous point, and we prove that a transitive system is either thickly sensitive or contains syndetically equicontinuous points, which is a refinement of another well-known result of Akin, Auslander and Berg.

title: On the Lyapunov numbers
abstract: We introduce and study the Lyapunov numbers -- quantitative measures of the sensitivity of a dynamical system $(X,f)$ given by a compact metric space $X$ and a continuous map $f:X \to X$. In particular, we prove that for a minimal topologically weakly mixing system all Lyapunov numbers are the same.

title: Sensitivity, proximal extension and higher order almost automorphy
abstract: Let $(X,T)$ be a topological dynamical system, and $\mathcal{F}$ be a family of subsets of $\mathbb{Z}_+$. $(X,T)$ is strongly $\mathcal{F}$-sensitive, if there is $\delta>0$ such that for each non-empty open subset $U$, there are $x,y\in U$ with $\{n\in\mathbb{Z}_+: d(T^nx,T^ny)>\delta\}\in\mathcal{F}$. Let $\mathcal{F}_t$ (resp. $\mathcal{F}_{ip}$, $\mathcal{F}_{fip}$) be consisting of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets). 
The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either strongly $\mathcal{F}_{fip}$-sensitive or an almost one-to-one extension of its $\infty$-step nilfactor. (2) a minimal system is either strongly $\mathcal{F}_{ip}$-sensitive or an almost one-to-one extension of its maximal distal factor. (3) a minimal system is either strongly $\mathcal{F}_{t}$-sensitive or a proximal extension of its maximal distal factor.

title: Disjointness and weak mixing of minimal sets
abstract: In [3, Problem G, p. 34] the following problem is proposed: characterize the class o' of all flows which are disjoint from every distal flow. We show here that ai consists precisely of the weakly mixing minimal flows. In arriving at this conclusion we make use of a result (Corollary to Theorem 1) which has been arrived at independently by Keynes and Robertson [4, Theorem 3.4, p. 366]. In the following all transformation groups will be assumed to have compact Hausdorff phase spaces. For any unexplained notation or terminology the reader is referred to  and . This research is part of the author's doctoral dissertation prepared at Yale University under the guidance of Professor S. Kakutani, whose assistance the author gratefully acknowledges. We say that a transformation group (X, T) is weakly mixing if given nonempty open subsets A, B, C, D of X there is tET such that AtnCC0 and BtnDS 0.

title: An explicit scattering, non-weakly mixing example and weak disjointness
abstract: By a dynamical system we mean a pair (X,T), where X is a compact metric space and T:X→X is surjective and continuous. We study weak disjointness in topological dynamics. (X,T) is scattering iff it is weakly disjoint from all minimal systems and (X,T) is strongly scattering iff it is weakly disjoint from all E-systems, i.e. transitive systems having invariant measures with full support. It is clear that a weakly mixing system is strongly scattering and the latter is scattering. An existential proof of scattering and a non-weakly mixing example is obtained by Akin and Glasner (2001 . In this paper, we will give an explicit example which is strongly scattering and not weakly mixing. We also define extreme scattering, weak scattering and study the relationships of the various definitions. For a dynamical property P stronger than transitivity, let P be the property such that a system has P iff it is weakly disjoint from any system having P. We show that P = P. Moreover, we prove that (thickly syndetic-transitive) = piecewise-syndetic-transitive and (piecewise-syndetic-transitive) = thickly syndetic-transitive.

title: On li-yorke pairs
abstract: The Li–Yorke definition of chaos proved its value for interval maps. In this paper it is considered in the setting of general topological dynamics. We adopt two opposite points of view. On the one hand sufficient conditions for Li–Yorke chaos in a topological dynamical system are given. We solve a long–standing open question by proving that positive entropy implies Li–Yorke chaos. On the other hand properties of dynamical systems without Li–Yorke pairs are investigated; in addition to having entropy 0, they are minimal when transitive, and the property is stable under factor maps, arbitrary products and inverse limits. Finally it is proved that minimal systems without Li–Yorke pairs are disjoint from scattering systems.

title: On weak mixing of cascades
abstract: The topological properties of "weak mixing" and "strong mixing" for topological transformation groups have been introduced and investigated and many analogous results have been derived. Let us adopt the following definitions and notations. We will restrict our attention to topological transformation groups (X, T) with compact metrizable phase space X. We call a topological transformation group ergodic if the only closed proper invariant subset is nowhere dense or, equivalently, there exists a point in X the orbit-closure of which is the whole space X. Such a point is called a transitive point. We call a topological transformation group weakly mixing if (X, T)x (X, T) is ergodic and strongly mixing if for any two open sets U and V, there is a compact subset K of T such that t ~ T ~ K implies Ut n V # ;J. Let ~'(X) be the set of all bounded complex-valued functions on X which are continuous on a co-meager set. (We identify functions which coincide on a co-meager set.) Let us define a cascade to be a topological transformation group with the discrete additive group of integers as phase group. In this situation, it is also denoted by (X, ~b), with ~b a homeomorphism of X. If Y is an invariant subset of X, then (Y, ~b) is also a cascade and we say that (Y, ~b) is a subcascade of (X, ~b). Let us consider the following properties of a topological transformation group (X, T):