(Redirected from Topologically transitive)
Then two points x and y in X will be topologically indistinguishable if the family does not separate them (i.e. = for all ). Given any equivalence relation on a set X there is a topology on X for which the notion of topological indistinguishability agrees with the given equivalence relation.
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc.
The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including strong mixing, weak mixing and topological mixing, with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a 'stronger' notion than ergodicity).
- 1Mixing in stochastic processes
- 2Mixing in dynamical systems
Mixing in stochastic processes[edit]
Let be a sequence of random variables. Such a sequence is naturally endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a sigma algebra, the Borel sigma algebra; it is the smallest (coarsest) sigma algebra that contains the topology.
Define a function , called the strong mixing coefficient, as
In this definition, P is the probability measure on the sigma algebra. The symbol , with denotes a subalgebra of the sigma algebra; it is the set of cylinder sets that are specified between times a and b. Given specific, fixed values , etc., of the random variable, at times , , etc., then it may be thought of as the sigma-algebra generated by
The process is strong mixing if as .
One way to describe this is that strong mixing implies that for any two possible states of the system (realizations of the random variable), when given a sufficient amount of time between the two states, the occurrence of the states is independent.
Types of mixing[edit]
Suppose {Xt} is a stationary Markov process, with stationary distribution Q. Denote the space of Borel-measurable functions that are square-integrable with respect to measure Q. Also let
denote the conditional expectation operator on Finally, let
denote the space of square-integrable functions with mean zero.
The ρ-mixing coefficients of the process {xt} are
The process is called ρ-mixing if these coefficients converge to zero as t → ∞, and “ρ-mixing with exponential decay rate” if ρt < e−δt for some δ > 0. For a stationary Markov process, the coefficients ρt may either decay at an exponential rate, or be always equal to one.[1]
The α-mixing coefficients of the process {xt} are
The process is called α-mixing if these coefficients converge to zero as t → ∞, it is “α-mixing with exponential decay rate” if αt < γe−δt for some δ > 0, and it is “α-mixing with sub-exponential decay rate” if αt < ξ(t) for some non-increasing function ξ(t) satisfying t−1ln ξ(t) → 0 as t → ∞.[1]
The α-mixing coefficients are always smaller than the ρ-mixing ones: αt ≤ ρt, therefore if the process is ρ-mixing, it will necessarily be α-mixing too. However when ρt = 1, the process may still be α-mixing, with sub-exponential decay rate.
The β-mixing coefficients are given by
The process is called β-mixing if these coefficients converge to zero as t → ∞, it is “β-mixing with exponential decay rate” if βt < γe−δt for some δ > 0, and it is “β-mixing with sub-exponential decay rate” if βtξ(t) → 0 as t → ∞ for some non-increasing function ξ(t) satisfying t−1ln ξ(t) → 0 as t → ∞.[1]
A strictly stationary Markov process is β-mixing if and only if it is an aperiodic recurrent Harris chain. The β-mixing coefficients are always bigger than the α-mixing ones, so if a process is β-mixing it will also be α-mixing. There is no direct relationship between β-mixing and ρ-mixing: neither of them implies the other.
Mixing in dynamical systems[edit]
A similar definition can be given using the vocabulary of measure-preserving dynamical systems. Let be a dynamical system, with T being the time-evolution or shift operator. The system is said to be strong mixing if, for any , one has
For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with replaced by with g being the continuous-time parameter.
To understand the above definition physically, consider a shaker full of an incompressible liquid, which consists of 20% wine and 80% water. If is the region originally occupied by the wine, then, for any part of the shaker, the percentage of wine in after repetitions of the act of stirring is
In such a situation, one would expect that after the liquid is sufficiently stirred (), every region of the shaker will contain approximately 20% wine. This leads to
where , because measure-preserving dynamical systems are defined on probability spaces, and hence the final expression implies the above definition of strong mixing.
A dynamical system is said to be weak mixing if one has
In other words, is strong mixing if in the usual sense, weak mixing if
in the Cesàro sense, and ergodic if in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing.
For a system that is weak mixing, the shift operatorT will have no (non-constant) square-integrableeigenfunctions with associated eigenvalue of one.[citation needed] In general, a shift operator will have a continuous spectrum, and thus will always have eigenfunctions that are generalized functions. However, for the system to be (at least) weak mixing, none of the eigenfunctions with associated eigenvalue of one can be square integrable.
formulation[edit]
The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical system is equivalent to the property that, for any function , the sequence converges strongly and in the sense of Cesàro to , i.e.,
A dynamical system is weakly mixing if, for any functions and
A dynamical system is strongly mixing if, for any function the sequence converges weakly to i.e., for any function
Since the system is assumed to be measure preserving, this last line is equivalent to saying that so that the random variables and become orthogonal as grows. Actually, since this works for any function one can informally see mixing as the property that the random variables and become independent as grows.
Products of dynamical systems[edit]
Given two measured dynamical systems and one can construct a dynamical system on the Cartesian product by defining We then have the following characterizations of weak mixing:
- Proposition. A dynamical system is weakly mixing if and only if, for any ergodic dynamical system , the system is also ergodic.
- Proposition. A dynamical system is weakly mixing if and only if is also ergodic. If this is the case, then is also weakly mixing.
Generalizations[edit]
The definition given above is sometimes called strong 2-mixing, to distinguish it from higher orders of mixing. A strong 3-mixing system may be defined as a system for which
holds for all measurable sets A, B, C. We can define strong k-mixing similarly. A system which is strongk-mixing for all k = 2,3,4,... is called mixing of all orders.
It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong m-mixing implies ergodicity.
Examples[edit]
Irrational rotations of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.
Many map considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature...
Topological mixing[edit]
A form of mixing may be defined without appeal to a measure, using only the topology of the system. A continuous map is said to be topologically transitive if, for every pair of non-empty open sets, there exists an integer n such that
where is the nth iterate of f. In the operator theory, a topologically transitive bounded linear operator (a continuous linear map on a topological vector space) is usually called hypercyclic operator. A related idea is expressed by the wandering set.
Lemma: If X is a completemetric space with no isolated point, then f is topologically transitive if and only if there exists a hypercyclic point, that is, a point x such that its orbit is dense in X.
A system is said to be topologically mixing if, given open sets and , there exists an integer N, such that, for all , one has
For a continuous-time system, is replaced by the flow, with g being the continuous parameter, with the requirement that a non-empty intersection hold for all .
A weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator.
Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.
References[edit]
- Chen, Xiaohong; Hansen, Lars Peter; Carrasco, Marine (2010). 'Nonlinearity and temporal dependence'. Journal of Econometrics. 155 (2): 155–169. CiteSeerX10.1.1.597.8777. doi:10.1016/j.jeconom.2009.10.001.
- Achim Klenke, Probability Theory, (2006) Springer ISBN978-1-84800-047-6
- V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (1968) W. A. Benjamin, Inc.
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(Redirected from Topologically distinguishable)
In topology, two points of a topological spaceX are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are points in X, and Nx is the set of all neighborhoods that contain x, and Ny is the set of all neighborhoods that contain y, then x and y are 'topologically indistinguishable' if and only ifNx = Ny.(See Hausdorff's axiomatic neighborhood systems.)
Intuitively, two points are topologically indistinguishable if the topology of X is unable to discern between the points.
Two points of X are topologically distinguishable if they are not topologically indistinguishable. This means there is an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely one of the two points). This open set can then be used to distinguish between the two points. A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. This is the weakest of the separation axioms.
Topological indistinguishability defines an equivalence relation on any topological space X. If x and y are points of X we write x ≡ y for 'x and y are topologically indistinguishable'. The equivalence class of x will be denoted by [x].
- 3Properties
Examples[edit]
For T0 spaces (in particular, for Hausdorff spaces) the notion of topological indistinguishability is trivial, so one must look to non-T0 spaces to find interesting examples. On the other hand, regularity and normality do not imply T0, so we can find examples with these properties. In fact, almost all of the examples given below are completely regular.
- In an indiscrete space, any two points are topologically indistinguishable.
- In a pseudometric space, two points are topologically indistinguishable if and only if the distance between them is zero.
- In a seminormed vector space, x ≡ y if and only if ‖x − y‖ = 0.
- For example, let L2(R) be the space of all measurable functions from R to R which are square integrable (see Lp space). Then two functions f and g in L2(R) are topologically indistinguishable if and only if they are equal almost everywhere.
- In a topological group, x ≡ y if and only if x−1y ∈ cl{e} where cl{e} is the closure of the trivial subgroup. The equivalence classes are just the cosets of cl{e} (which is always a normal subgroup).
- Uniform spaces generalize both pseudometric spaces and topological groups. In a uniform space, x ≡ y if and only if the pair (x, y) belongs to every entourage. The intersection of all the entourages is an equivalence relation on X which is just that of topological indistinguishability.
- Let X have the initial topology with respect to a family of functions . Then two points x and y in X will be topologically indistinguishable if the family does not separate them (i.e. for all ).
- Given any equivalence relation on a set X there is a topology on X for which the notion of topological indistinguishability agrees with the given equivalence relation. One can simply take the equivalence classes as a base for the topology. This is called the partition topology on X.
Specialization preorder[edit]
The topological indistinguishability relation on a space X can be recovered from a natural preorder on X called the specialization preorder. For points x and y in X this preorder is defined by
- x ≤ yif and only ifx ∈ cl{y}
where cl{y} denotes the closure of {y}. Equivalently, x ≤ y if the neighborhood system of x, denoted Nx, is contained in the neighborhood system of y:
- x ≤ y if and only if Nx ⊂ Ny.
It is easy to see that this relation on X is reflexive and transitive and so defines a preorder. In general, however, this preorder will not be antisymmetric. Indeed, the equivalence relation determined by ≤ is precisely that of topological indistinguishability:
- x ≡ y if and only if x ≤ y and y ≤ x.
A topological space is said to be symmetric (or R0) if the specialization preorder is symmetric (i.e. x ≤ y implies y ≤ x). In this case, the relations ≤ and ≡ are identical. Topological indistinguishability is better behaved in these spaces and easier to understand. Note that this class of spaces includes all regular and completely regular spaces.
Properties[edit]
Equivalent conditions[edit]
There are several equivalent ways of determining when two points are topologically indistinguishable. Let X be a topological space and let x and y be points of X. Denote the respective closures of x and y by cl{x} and cl{y}, and the respective neighborhood systems by Nx and Ny. Then the following statements are equivalent:
- x ≡ y
- for each open set U in X, U contains either both x and y or neither of them
- Nx = Ny
- x ∈ cl{y} and y ∈ cl{x}
- cl{x} = cl{y}
- x ∈ ∩Ny and y ∈ ∩Nx
- ∩Nx = ∩Ny
- x ∈ cl{y} and x ∈ ∩Ny
- x belongs to every open set and every closed set containing y
- a net or filter converges to x if and only if it converges to y
These conditions can be simplified in the case where X is symmetric space. For these spaces (in particular, for regular spaces), the following statements are equivalent:
- x ≡ y
- for each open set U, if x ∈ U then y ∈ U
- Nx ⊂ Ny
- x ∈ cl{y}
- x ∈ ∩Ny
- x belongs to every closed set containing y
- x belongs to every open set containing y
- every net or filter that converges to x converges to y
Equivalence classes[edit]
To discuss the equivalence class of x, it is convenient to first define the upper and lower sets of x. These are both defined with respect to the specialization preorder discussed above.
The lower set of x is just the closure of {x}:
while the upper set of x is the intersection of the neighborhood system at x:
The equivalence class of x is then given by the intersection
![If F Is Topologically Transitive Then F^n Is Transitive If F Is Topologically Transitive Then F^n Is Transitive](https://i1.rgstatic.net/publication/265718808_The_Asymptotic_Average_Shadowing_Property_and_Its_Applications/links/54198f640cf2218008bf8ea9/largepreview.png)
Since ↓x is the intersection of all the closed sets containing x and ↑x is the intersection of all the open sets containing x, the equivalence class [x] is the intersection of all the open and closed sets containing x.
Both cl{x} and ∩Nx will contain the equivalence class [x]. In general, both sets will contain additional points as well. In symmetric spaces (in particular, in regular spaces) however, the three sets coincide:
In general, the equivalence classes [x] will be closed if and only if the space is symmetric.
Continuous functions[edit]
Let f : X → Y be a continuous function. Then for any x and y in X
- x ≡ y implies f(x) ≡ f(y).
The converse is generally false (There are quotients of T0 spaces which are trivial). The converse will hold if X has the initial topology induced by f. More generally, if X has the initial topology induced by a family of maps then
- x ≡ y if and only if fα(x) ≡ fα(y) for all α.
It follows that two elements in a product space are topologically indistinguishable if and only if each of their components are topologically indistinguishable.
Kolmogorov quotient[edit]
Since topological indistinguishability is an equivalence relation on any topological space X, we can form the quotient spaceKX = X/≡. The space KX is called the Kolmogorov quotient or T0 identification of X. The space KX is, in fact, T0 (i.e. all points are topologically distinguishable). Moreover, by the characteristic property of the quotient map any continuous map f : X → Y from X to a T0 space factors through the quotient map q : X → KX.
Although the quotient map q is generally not a homeomorphism (since it is not generally injective), it does induce a bijection between the topology on X and the topology on KX. Intuitively, the Kolmogorov quotient does not alter the topology of a space. It just reduces the point set until points become topologically distinguishable.
See also[edit]
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