On -Compatible Metrics and Measurable Sensitivity
We introduce the notion of W-measurable sensitivity, which extends and strictly implies canonical measurable sensitivity, a measure-theoretic version of sensitive dependence on initial conditions. This notion also implies pairwise sensitivity with respect to a large class of metrics. We show that nonsingular ergodic and conservative dynamical systems on standard spaces must be either W-measurably sensitive, or isomorphic mod 0 to a minimal uniformly rigid isometry. In the finite measure-preserving case they are W-measurably sensitive or measurably isomorphic to an ergodic isometry on a compact metric space.
The notion of sensitive dependence on initial conditions is an extensively studied isomorphism invariant of topological dynamical systems on compact metric spaces ([GW93], [AAB96]). In [JKL08], the authors define two measure-theoretic versions of sensitive dependence, measurable sensitivity and strong measurable sensitivity, and show that, unlike their traditional topologically-dependent counterpart, both of these properties carry up to measurable-theoretic isomorphism. James et. al. introduce these notions for nonsingular transformations and show that measurable sensitivity is implied by double ergodicity (a property equivalent to weak mixing in the finite measure-preserving case) and strong measurable sensitivity is implied by light mixing in the finite measure-preserving case.
In this paper, we introduce W-measurable sensitivity, a notion that is a priori stronger than measurable sensitivity and implies it straightforwardly. We use this new property, together with properties of -compatible metrics (see below), to formulate a classification of all nonsingular conservative and ergodic transformations on standard Borel spaces as being either W-measurably sensitive or isomorphic to a minimal uniformly rigid isometry; in the case of finite invariant measure we obtain more, namely W-measurably sensitive or isomorphic to a minimal uniformly rigid invertible isometry on a compact metric space. In the course of this proof, we also show that W-measurable sensitivity is in fact equivalent to measurable sensitivity for conservative and ergodic transformations.
In addition, we show (see Appendix A) that the notion of W-measurable sensitivity is closely related to pairwise sensitivity, a notion introduced in [CJ05] for finite measure-preserving transformations. In their paper, Cadre and Jacob show that weakly mixing finite measure-preserving transformations always exhibit pairwise sensitivity, and also any ergodic finite measure-preserving transformation satisfying a certain entropy condition. Our results imply that any finite measure-preserving ergodic transformation that is not isomorphic mod 0 to a Kronecker transformation will exhibit pairwise sensitivity with respect to any -compatible metric (in addition to W-measurable sensitivity).
The plan of the paper is as follows. Section 2 recalls basic definitions from [JKL08] and introduces -compatible metrics and some of their properties. In Section 3 we define W-measurable sensitivity. Section 4 starts by construncting -Lipshitz metrics from any metric on a dynamical system, and then shows that W-measurable sensitivity can be equivalently expressed in additional ways using properties of -compatible metrics. In Section 5, we provide a sufficient condition under which the newly constructed -Lipshitz metric is in fact -compatible, and discuss consequences of this fact largely from [AG01]. In Section 6 we discuss the invariance of W-measurable sensitivity under measurable isomorphism, as well as the technical assumptions necessary for it to hold. We also illustrate the main connection between -Lipshitz metrics and W-measurable sensitivity, namely that a conservative and ergodic nonsingular dynamical system is W-measurably sensitive if and only if all dynamical systems isomorphic mod 0 to it admit no -compatible -Lipshitz metrics. Finally, in Section 7 we prove our main result, which classifies all conservative and ergodic, nonsingular transformations on standard Borel spaces as being either W-measurably sensitive, or isomorphic to a minimal uniformly rigid invertible isometry. A corollary of this fact is that for conservative and ergodic transformations, W-measurable sensitivity is equivalent to measurable sensitivity as defined in [JKL08]. We end the section by obtaining a stronger result in the case of ergodic finite measure-preserving transformations.
This paper is based on research by the Ergodic Theory group of the 2007 SMALL summer research project at Williams College. Support for the project was provided by National Science Foundation REU Grant DMS - 0353634 and the Bronfman Science Center of Williams College. The first-named author would also like to acknowledge support by an NSF graduate fellowship.
We are indebted to the referee for a careful reading of the manuscript and several comments and suggestions that improved our paper. We thank Ethan Akin for several remarks including an argument that removed the assumption of forward measurability in an earlier version of our paper, the proof of Proposition 5.6, and for bringing [HLY] to our attention.
2. Preliminary Definitions
A nonsingular dynamical system is a quadruple , where is a standard nonatomic Lebesgue space (i.e., is a standard Borel space, see e.g. [Sri98], and is a -finite, nonatomic measure on ). It follows that must be of cardinality as the measure is nonatomic. Furthermore, the transformation is measurable and a nonsingular endomorphism (i.e., for all , and if and only if , see e.g. [Sil08]). In some cases we assume that is measure-preserving or that the measure space is finite. Recall that is conservative and ergodic if and only if for all measurable sets , if , then or .
We consider metrics or pseudo-metrics on . We assume throughout this article that all pseudo-metrics are (Borel) measurable and bounded by (one can replace by ). It follows that, for each , the set is measurable. Therefore, by e.g. [Sri98, Exercise 3.1.20], the balls
are measurable. For a pseudo-metric define
A (measurable) metric on is said to be -compatible if assigns positive (nonzero) measure to all nonempty, open -balls in , equivalently if , or if for all . If a -compatible metric on , then is separable under (see [JKL08, 1.1] and Proposition 2.1 below). Therefore open sets are measurable as they are countable unions of balls. All -closed sets are also measurable, etc. We say that is -separable if , or equivalently a.e. If follows that if is -separable, then the restriction of to is -compatible.
Let be a nonsingular dynamical system and let be a pseudo-metric on .
The function is continuous with respect to and measurable.
The pseudo-metric is separable when restricted to . In particular, if is -compatible, then it is separable on .
is open with respect to and measurable.
A pseudo-metric is -separable if and only if there exists a measure zero subset of such that restricted to is separable.
(1) Suppose that . Set
The for each we have
Since , , so and . Similarly we obtain that . This implies that is continuous with respect to , and therefore measurable.
(2) For , let be such that if then and let it be maximal with respect to this property. It follows that
is a collection of disjoint sets of positive measure and since is -finite, this collection is countable. This shows that each is countable. Then the union , for , is a countable set that is dense in for the metric .
(3) Since is continuous by part (1), is open with respect to . By part (2), every open set that is contained in is a countable union of balls, hence it is measurable. Similarly, closed sets contained in are measurable. In particular, , and so , are measurable.
(4) Suppose that and let be such that . We show that is not separable on the subset of . We first note that the collection
is an open cover of , and since has positive measure and each of the balls has measure zero (by definition of ), the collection cannot have a countable subcover. Conversely, if we can let and use part (2). ∎
Let be a nonsingular dynamical system and let be a pseudo-metric on . Let . If for almost all , then for all .
We know that . Suppose for some . Then . So there exists . By the triangle inequality, . This means that , a contradiction. ∎
3. W-measurable Sensitivity
We start by recalling the definition of measurable sensitivity.
[JKL08] A nonsingular dynamical system is said to be measurably sensitive if for every isomorphic mod dynamical system and any -compatible metric on , then there exists such that for and all there exists such that
We now introduce the definition that we shall be using extensively.
For a -compatible metric , a nonsingular dynamical system is W-measurably sensitive with respect to if there is a such that for every ,
for almost every . The dynamical system is said to be W-measurably sensitive if the above definition holds true for all -compatible metrics .
Remark. (1) As in [JKL08], it can be shown that a doubly ergodic nonsingular transformation is W-measurably sensitive. (Double ergodicity is a condition for nonsingular transformations that is equivalent to weak mixing in the finite measure-preserving case [Fur81].) There exist both infinite (and finite) measure-preserving and nonsingular type III (i.e., not admitting an equivalent -finite invariant measure) invertible transformations that are doubly ergodic (see e.g. [DS09]), and therefore W-measurably sensitive.
(2) If a measure space has atoms, no transformation on it can exhibit W-measurable sensitivity with respect to any metric. Indeed, for any , and any , the set of points such that cannot include . So this set cannot have full measure (i.e., its complement has measure zero) if .
The same is not true about measurable sensitivity. For this reason, throughout this paper we assume that our measure space is nonatomic.
(3) A very important example of an ergodic finite measure-preserving dynamical system which is not W-measurably sensitive is a Kronecker transformation, i.e. an ergodic isometry on an interval of finite length (with the Lebesgue measure and the usual metric). This transformation is not W-measurably sensitive with respect to the usual metric because it is an isometry. There are also examples of conservative and ergodic type III nonsingular invertible transformations that are not W-measurable sensitive. Let , the -adic integers, let addition by 1, , and be the -adic metric. Then it is well known that is a minimal isometry for . Let and , a probability measure on the Borel -field . Then is a nonsingular measure for that is conservative and ergodic of type III (when ), see e.g. [DS09]. It is clear that is -compatible, so is a conservative ergodic invertible nonsingular transformation that is not finite measure-preserving and is not W-measurably sensitive.
We note that, the property of W-measurable sensitivity is preserved under measurable isomorphisms (Proposition 6.2).
W-measurable sensitivity clearly implies measurable sensitivity (see first part of the proof of Proposition 7.2). In fact, we show that the two notions are equivalent for conservative and ergodic dynamical systems. We first show in Proposition 4.2 that for a transformation to be W-measurably sensitive, it is sufficient for each to have one value of that satisfies . The remainder of the equivalence follows from the results in the following sections, culminating with Proposition 7.2.
4. Constructing -Lipshitz Metrics
We shall use the term -Lipshitz metrics (with respect to ) to denote metrics that satisfy the inequality for all and .
First, we provide a way to construct a -Lipshitz metric from any other metric.
Let be a nonsingular dynamical system, and be a metric on . Define, for ,
is a metric on (satisfying our standing assumptions: measurable and bounded). Moreover, it is a -Lipshitz metric.
The first statement is left to the reader. To see that it is -Lipshitz we compute,
Remark. In general, even if the metric is -compatible, the metric may not be -compatible. Consequently, there is no guarantee that the measure space is separable under the topology determined by .
For example, let be the unit interval, be the Lebesgue measure, and be the usual metric. Let be the doubling map . Note that is a -compatible metric.
The metric , however, is not -compatible. Indeed, for any , and any , there will be an such that . So, since , we have
In other words, for any , the ball around 0 in the metric may contain only rational points. So, , and is not -compatible.
In this example, the transformation turns out to be W-measurably sensitive. In fact, since mixing, it is strongly measurably sensitive (see [JKL08]). On the other hand, we will see that whenever the -Lipshitz metric is -compatible, the corresponding transformation is not W-measurably sensitive.
We now formulate several equivalent definitions of W-measurably sensitive transformations. We start by showing that while the original definition requires the existence of infinitely many times satisfying the condition, it is sufficient to require the existence of one such .
Let be a nonsingular dynamical system, and be a -compatible metric. The following are equivalent:
The system is W-measurably sensitive with respect to .
There is a such that, for each , for almost every ,
There is a such that for each ,
There is a such that for each ,
There is a such that for each ,
. Suppose that there is a such that for each , for almost every , there exists such that . For every natural number and define a set by:
We now prove that for all and , the set has full measure. Consider the point . Using our assumption, for almost every , there exists such that . In other words, the set
has full measure. Notice that . Since is a nonsingular transformation, must also have full measure.
Finally, let . Clearly, has full measure. Furthermore, for every , there are infinitely many values of such that . So
for almost all . Therefore the system is W-measurably sensitive with respect to .
. The converse is clear from the definitions.
. If condition is satisfied at for some , then is contained in the complement of a set of full measure. So .
Conversely, if condition is satisfied at for some , then has measure zero. So in particular, the set has measure zero. Therefore, for almost every , there is some for which , and condition is satisfied.
The equivalence of and is clear form the definitions. The equivalence of and is clear since does not have to be the same. ∎
5. Conditions for -Lipshitz metric to be -compatible and Consequences
Now, we provide a sufficient condition for the -Lipshitz metric to be -compatible given that the transformation is ergodic.
The proof of the following lemma is standard, see for example [ST91, Corollary 2.7].
Let be a conservative and ergodic nonsingular dynamical system. Let be a measurable function. If a.e., then a.e.
Let be a nonsingular dynamical system, and be a metric on . If is –Lipshitz then
Let denote the metric . First we observe
Since is nonsingular, if and only if . It follows that
Since is -Lipshitz, , which implies
completing the proof. ∎
Now, we are ready to state the sufficient condition the -Lipshitz metric to be -compatible which is our main tool in proving the main results in Section 7.
Let be a conservative and ergodic nonsingular dynamical system. Let be a -compatible metric on . Suppose further that is not W-measurably sensitive with respect to . Then there exists a positively invariant measurable set of full measure (i.e., and ) such that is a -compatible metric for the system , where and are the restrictions to of the original measure and transformation.
First we observe that
In fact, if , then . Since is -Lipschitz, by Lemma 5.2, , so . Therefore can be restricted to a transformation on the positively invariant set .
Since is conservative and ergodic it follows from (1) that or . If it were the case that then there would exist such that on a set of positive measure, hence by Lemmas 5.2 and 5.1, as is conservative and ergodic, the condition holds for a.e. , but this contradicts the hypothesis by the Remark following Proposition 4.2. Therefore and is a set of full measure. (It follows also that )
Clearly, is a metric on . To see that it is -compatible we calculate, for and ,
In relation to Lemma 5.3 , we note that it is possible that a system is not W-measurably sensitive, but does not itself admit any -compatible metric that is -Lipschitz. For example, consider the dynamical system where is the unit interval and is the Lebesgue measure. Let be a fixed irrational number between 0 and 1. For any , we define:
This system is ergodic and not measurably sensitive as it is measurably isomorphic to a rotation. However, there is no -compatible -Lipshitz metric on .
Indeed, suppose that there is a -compatible metric such that for all . Let be a ball of radius around 0. Since is -compatible, must have positive measure. Furthermore, since , for any point , we must have and, therefore, . So maps a set of positive measure into itself. This is impossible for a transformation isomorphic mod 0 to an irrational rotation.
In the rest of this section, we describe some useful consequences of a 1-Lipshitz metric being -compatible.
Let be a metric space and a transformation. Let denote the set of accumulation points of the positive orbit . A point is a transitive point for if . When has no isolated points this is equivalent to the (positive) orbit of being dense in . As we will only consider -compatible metrics where is nonatomic, all our metric spaces will have no isolated points. is transitive if it has a transitive point. The transformation is minimal if for all . It is uniformly rigid if there exists a sequence such that converges to uniformly on .
The following lemma is essentially known.
Let be a conservative and ergodic nonsingular dynamical system. If is a -compatible metric on , then -a.e. point of is transitive.
Since by assumption is nonatomic, has no isolated points. By Proposition 2.1, is separable, so there exist dense in . For each and each , set
Since is conservative and ergodic, each is of full measure. Finally let
Clearly is of full measure and each point in has a dense orbit. ∎
The following proposition is essentially from [AG01].
Let be a metric space and let be a -Lipschitz transformation. If is transitive, then it is a uniformly rigid, minimal isometry.
Let be a point such that . (This in particular implies that the metric is separable.) Let . There exists an integer such that . Since is -Lipschitz, for all , . Let . Since is continuous, for such that is sufficiently small, . Then
Therefore is uniformly rigid. Now, in this case there exists a sequence such that for all . Therefore, for all ,
If were not an isometry there would exist , such that , but then could not converge to .
Finally we show that is minimal. Again, let and . Let , . There exists such that . Then we can choose so that . Then
Therefore . ∎
Now, let be the space of continuous maps from to itself, with the metric . We also define a subset
This is clearly a sub-semigroup of under composition.
The following proposition is essentially from [AG01]. We are indebted to Ethan Akin for the proof.
Let be a metric space and let be a transitive and -Lipshitz transformation. Then, for each , the evaluation map
is an isometry. Also, the space is the closure of sequence in . If in addition the metric space is complete, then the evaluation map is an invertible isometry. Moreover, the semigroup is then a group, and therefore has to be invertible.
Fix a point and let . We wish to show that the map is an isometry. Since and both commute with , and is -Lipshitz, for all ,
Since and are both continuous and the set of all is dense, for all , and therefore
and so is an isometry.
Now, the subset is clearly closed in . Fix some and . Since is minimal, is a transitive point, and so there is a sequence such that . In other words, in . Since is an isometry, this implies that in , completing the proof of the first part of the proposition.
If we assume that the space is complete, so is the space . For , we show that is surjective.
Pick a . There is a sequence of -s such that . In particular, the sequence is Cauchy. Since is an isometry, the sequence is Cauchy in . By completeness, it has a limit (since is closed); clearly and is surjective.
Now, let be arbitrary. Since the map is surjective, we can pick an so that
Since and is injective, is the identity, and . So, all maps in are invertible. ∎
6. W-measurable sensitivity on isomorphic mod 0 dynamical systems
We prove that W-measurable sensitivity is invariant under measurable isomorphism. Here we use that we are working on standard Borel spaces.
Let be a standard Borel space, with a nonatomic measure on . Let be a Borel subset of full measure and let be a -compatible metric defined on . Then the metric can be extended to a -compatible metric on all of in such a way that and agree on a set of full measure.
Since the measure is nonatomic and is Borel, it must have the same cardinality as . Using e.g. [Sri98, 3.4.23] one can show that there exists a Borel set of measure zero and cardinality . Therefore there exists a Borel isomorphism . Then we can define by
( is the identity on the full-measure Borel subset .) For define . Clearly, since is a measurable metric, so is . Since every -ball corresponds to a -ball under the map , which is a Borel isomorphism, is also a -compatible metric and agrees with on . ∎
Using Lemma 6.1, we can prove the invariance of W-measurable sensitivity.
Suppose is a W-measurably sensitive nonsingular dynamical system. Let be a nonsingular dynamical system isomorphic mod 0 to . Then, is also W-measurably sensitive.
Suppose is not W-measurably sensitive. Then, there is a -compatible metric on such that is not W-measurably sensitive with respect to .
By the definition of measurable isomorphism, there must be Borel subsets and and a measure-preserving bijection such that , and .
We define a metric on by for . It is clearly -compatible on . We apply Lemma 6.1 to extend to a -compatible metric defined on all of that agrees with almost everywhere.
Now, we show that is not W-measurably sensitive with respect to . Let . Since is not W-measurably sensitive with respect to , by part of Proposition 4.2, there must be an such that the set has positive measure. Let be the corresponding set in , that is . Note that .
Pick any . By the triangle inequality, for all and all integers , we have:
Since has positive measure, cannot be W-measurably sensitive. ∎
Let be a conservative and ergodic nonsingular dynamical system. is W-measurably sensitive if and only if all measurably isomorphic dynamical systems admit no -compatible metrics that are -Lipshitz.
First we note that if a dynamical system admits a -compatible -Lipshitz metric , then this system could not be W-measurably sensitive, since for all integers , . Now, if a dynamical system is W-measurably sensitive, then every measurably isomorphic system will also be W-measurably sensitive, and therefore will not admit a -compatible -Lipshitz metric .
For the converse, suppose is not W-measurably sensitive. By Lemma 5.3 there is a set of full measure , such that if is restricted to and to be restricted to , then is a -Lipshitz -compatible metric on . ∎
Remark. If is a -compatible metric on , must be a separable metric space under [JKL08] and Proposition[(3)] 2.1, so has at most the cardinality of the reals. A nonatomic (probability) Lebesgue space is defined as a measure space that is isomorphic mod 0 to the unit interval with Lebesgue measure , i.e., there exists sets of full measure and such that there is a (measure-preserving) isomorphism from to . However, there is not restriction on other than it is of -measure 0 and it could have cardinality greater than the reals. In this case would admit no -compatible metric, and for instance, transformations on this space would be vacuously W-measurably sensitive. We introduce the following definition for Lebesgue spaces.
Let be a Lebesgue space (or more generally a -finite measure space) and let be a nonsingular transformation on . A dynamical system is VW-measurably sensitive if for every positively invariant measurable set of full measure set , the system is W-measurably sensitive.
Remark. (1) By Lemma 6.1, on standard Borel spaces, the notions of W-measurable sensitivity and VW-measurable sensitivity are equivalent. Also, it follows from the definition that VW-measurable sensitivity is invariant under isomorphism.
(2) Here we note that nonsingular dynamical system (on standard Borel spaces) do admit -compatible measures. If fact we know that if is a standard Borel space and is a continuous measure on , which we may assume a probability measure, then there exists a Borel isomorphism from to the unit interval with Lebesgue measure (see e.g. [Sri98, 3.4.23]. Clearly Euclidean distance on is a -compatible measure on . Then defined by is a -compatible metric on .
7. Characterization of W-measurable Sensitivity
We shall prove our main result, that such a transformation is either W-measurably sensitive or measurably isomorphic to a minimal uniformly rigid isometry. This can be seen as a measurable version of the Dichotomy Theorem of Auslander and Yorke [AY80] for topological dynamical systems (continuous surjective maps on compact metric spaces), which states that a transitive map on a topological system is either sensitive or almost equicontinuous. Related topological dynamical results are in [GW93], [AG01] and the references therein.
Let be a conservative and ergodic nonsingular dynamical system. Then is either W-measurably sensitive or is isomorphic mod 0 to an invertible minimal uniformly rigid isometry on a Polish space.
Suppose is not W-measurably sensitive. Then, by Lemma 5.3, there exists a positively invariant set of full measure such that is -compatible for the system , where is the restriction of to and the restriction of to . By Lemma 5.4, is transitive with respect to . Since is 1-Lipshitz with respect to , by Proposition 5.5, is a uniformly rigid minimal isometry on .
Now, let be the topological completion of the metric space . Since is separable, is also separable so is Polish. We extend the measure to by defining a set to be measurable if is measurable, with . Since is an isometry, it is continuous on , so there is a unique way to extend it to a continuous transformation on . It’s easy to verify that must also be an isometry with respect to . It is invertible by Proposition 5.6.
Clearly, the dynamical system is measurably isomorphic to . ∎
Invertible examples of W-measurably sensitive transformations are mentioned in Section 3, but we have the following direct consequence of the theorem.
If a conservative and ergodic nonsingular transformation is not invertible a.e. then it cannot be isomorphic mod 0 to an invertible isometry, so it must be W-measurably sensitive.
As a first application of Theorem 1, we show the following proposition.
If a dynamical system is W-measurably sensitive, then it is measurably sensitive. If a dynamical system is conservative ergodic and measurably sensitive, then it is W-measurably sensitive.
First, suppose is a W-measurably sensitive nonsingular dynamical system. By Proposition 6.2, every isomorphic mod 0 dynamical system is also W-measurably sensitive. So, for any -compatible metric on , there is a such that for all , we have for almost all .
This implies that there is an for which the set
has positive measure. Thus is measurably sensitive.
To show the convere, suppose is a conservative and ergodic dynamical system that is not W-measurably sensitive. Then, by Theorem 1, there is a isomorphic mod 0 dynamical system and a -compatible metric on that is an isometry. For all , choose any , and then for any with , for all integers ,