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# [

Sophie Grivaux CNRS, Laboratoire Paul Painlevé, UMR 8524, Université Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France    Étienne Matheron Laboratoire de Mathématiques de Lens, Université d’Artois, rue Jean Souvraz S. P. 18, 62307 Lens (France).
August 28, 2019
###### Abstract

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for ; that there exist frequently hypercyclic operators on the sequence space admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are are satisfied by all the known chaotic operators) for an operator to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support.

Linear dynamical systems, frequently hypercyclic operators, chaotic operators, invariant and ergodic measures

Invariant measures for frequently hypercyclic operators]Invariant measures for
frequently hypercyclic operators

\subjclass

47A16, 47A35, 37A05

thanks: We are grateful to B. Weiss for pointing out to us the example mentioned in Section 1.2.

## 1 Introduction and main results

### 1.1 General background

Let be a separable, infinite-dimensional Banach space, and let us denote by the set of all continuous linear operators on . If , the pair is called a linear dynamical system. For every , we denote by the orbit of under the action of ,

 Orb(x,T)={Tnx;n∈N}.

A linear dynamical system is a special case of a Polish dynamical system, i.e. a continuous map acting on a Polish space. As such, it can be studied from different points of view.

- One can adopt a purely topological viewpoint, investigating in particular the individual behaviour of orbits. A basic notion in this context is that of hypercyclicity: the operator is said to be hypercyclic if there exists a vector , called a hypercyclic vector for , whose orbit is dense in . It is well-known that is hypercyclic if and only if it is topologically transitive, i.e. for each pair of nonempty open sets in , one can find such that ; and in this case, the set of hypercyclic vectors for , denoted by , is a dense subset of .

- One can also adopt a measure-theoretic point of view and try to find conditions on , and possibly on , ensuring that admits an invariant measure with some interesting property. (Note that since , the Dirac mass is an invariant measure for , arguably not very interesting). For example, one may look for an invariant measure with full support (i.e. such that for every open set ), or even an invariant measure with full support with respect to which has some ergodicity property, e.g. ergodicity in the usual sense, weak mixing or strong mixing. This kind of questions goes back at least to the classical work of Oxtoby and Ulam [23].

All measures considered in this paper will be finite Borel measures, and we will most of the time consider only probability measures without writing explicitly so. We will denote by the set of all -invariant Borel probability measures on .

The study of ergodic properties of linear dynamical systems was launched by Flytzanis in the paper [14] and pursued in [2], leading to the obtention of necessary and sufficient conditions for operators on complex Hilbert spaces to admit Gaussian invariant measures with one of the above properties. (A Borel probability measure on a complex Banach space is said to be Gaussian if every continuous linear functional has a complex Gaussian distribution when considered as a random variable on ; see e.g [4, Chapter 5] for more on Gaussian measures). The quest for similar results for operators acting on general Banach spaces was begun in [3] and culminated in [5], where a very general condition (valid on any complex Banach space ) for an operator to admit a Gaussian ergodic measure with full support was obtained.

Since we shall refer to this result from [5] below and use some relevant terminology, we now state it precisely. Assume that is a complex Banach space. By a unimodular eigenvector for an operator , we mean an eigenvector whose associated eigenvalue has modulus . We say that has a perfectly spanning set of unimodular eigenvectors if, for every countable set , we have

 ¯¯¯¯¯¯¯¯¯¯¯span[ker(T−λI);λ∈T∖D]=X.

This notion was introduced in [14] in a Hilbert space setting. A formally stronger property was considered in [2], and it was shown later on in [16] that the two notions are in fact equivalent. The aforementioned result from [5] states that any operator with a perfectly spanning set of unimodular eigenvectors admits a Gaussian ergodic measure with full support. Without the requirement that the measure should be Gaussian, this was essentially proved earlier in [16].

We refer the reader to the book [4] for more about linear dynamical systems, both from the topological and the ergodic theoretical points of view. Other interesting references are the book [19] (which is more concerned with hypercyclicity issues), and the recent paper [13] for its point of view on linear systems as special cases of Polish dynamical systems.

In the present paper, we shall consider measure-theoretic properties of linear operators outside of the Gaussian framework.

Our main interest lies with the so-called frequently hypercyclic operators. Frequent hypercyclicity, which was introduced in [2], is a strengthening of hypercyclicity which quantifies the frequency with which the orbit of a vector visits a given nonempty open set.

The precise definition reads as follows. Let . For any set and any , let us set

 NT(x,B):={n∈N;Tnx∈B}.

The operator is said to be frequently hypercyclic if there exists a vector such that for each nonempty open set , the set of integers has positive lower density; in other words,

 liminfN→∞1N#{n∈[1,N];Tnx∈V}>0.

Such a vector is said to be a frequently hypercyclic vector for , and the set of all frequently hypercyclic vectors is denoted by .

We have defined frequent hypercyclicity for linear dynamical systems only, but the notion obviously makes sense for an arbitrary Polish dynamical system . In fact, in the compact case such systems were studied long before frequently hypercyclic operators – see e.g. Furstenberg’s book [15]. Likewise, what could be called hypercyclic systems (i.e. Polish dynamical systems admitting dense orbits) are of course central objects in topological dynamics. In this paper, we will use the linear terminology (hypercyclic, frequently hypercyclic) for both linear dynamical systems and general Polish dynamical systems.

### 1.2 Two basic questions

One of the interests of frequent hypercyclicity is that, although its definition is purely topological, its is naturally and deeply linked to measure-theoretic considerations about the dynamical system . This is for instance testified by the following two facts.

- If is a Polish dynamical system and if admits an ergodic measure with full support, then is frequently hypercyclic and has full -measure. This follows easily from the pointwise ergodic theorem.

- If is a compact dynamical system and if is frequently hypercyclic, then admits an invariant probability measure with full support. A proof of this statement can be found e.g. in [15]; it will also be briefly recalled at the beginning of Section 2.

These observations make it natural to wonder whether an arbitrary frequently hypercyclic Polish dynamical system always admits an invariant probability measure with full support. That the answer to this question is negative was kindly pointed out to us by B. Weiss. Let be any irrational rotation of , and let be the normalized Lebesgue measure on . Then is a uniquely ergodic dynamical system whose unique invariant probability measure is and for which all points are frequently hypercyclic. Let be a nowhere dense compact subset of such that , and set . Then is a -invariant, dense subset of , and hence is a frequently hypercyclic Polish dynamical system. On the other hand, we have because and is an ergodic measure for . Since is uniquely ergodic, it follows that admits no invariant measure at all.

This example is however highly non linear, and one may naturally ask what happens in the linear setting.

###### Question 1.1.

Let be a frequently hypercyclic linear dynamical system. Is it true that admits an invariant probability measure with full support?

One may also ask whether a stronger result holds true:

###### Question 1.2.

Let be a frequently hypercyclic linear dynamical system. Is it true that admits an ergodic probability measure with full support?

Our first main result is concerned with Question 1.1. We obtain a positive answer when is a linear operator acting on a reflexive Banach space .

Recall that a measure on is said to be continuous if for every . Denoting by the set of all periodic points of , it is not hard to see that if is a -invariant measure such that , then is necessarily continuous.

###### Theorem 1.3.

If is a reflexive Banach space, then any frequently hypercyclic operator on admits a continuous invariant probability measure with full support. In fact, one may require that the measure satisfies .

Applying the ergodic decomposition theorem (see e.g [1, Ch. , Sec. ]), we easily deduce the following result:

###### Corollary 1.4.

If is a reflexive Banach space and if is a nonempty open set, then any frequently hypercyclic operator on admits a continuous ergodic probability measure such that .

Indeed, let be an invariant measure with full support for such that . By the ergodic decomposition theorem, one may write

 m=∫Sμsdp(s),

where the are ergodic probability measures for and the integral is taken over some probability space . The meaning of the above formula is that for every Borel set . As has full support, we have and hence ; and since we have for -almost every . So one can find such that the measure has the required properties.

The existence of an invariant measure with full support in Theorem 1.3 relies on the following “nonlinear” statement, which is valid for a rather large class of Polish dynamical systems.

###### Theorem 1.5.

Let be a Polish dynamical system. Assume that is endowed with a topology which is Hausdorff, coarser than the original topology , and such that every point of has a neighbourhood basis with respect to consisting of -compact sets. Moreover, assume that

• is frequently hypercyclic with respect to ;

• is a continuous self-map of .

Then admits an invariant probability measure with full support.

Forgetting the continuity requirement on , the first part of Theorem 1.3 follows at once from this result, taking for the weak topology of the reflexive Banach space and using the fact that the closed balls of are weakly compact.

Since compactness is rather crucial in all our arguments, and although we are not able to provide a counterexample, it seems reasonable to believe that Theorem 1.3 breaks down when the space is not assumed to be reflexive. Good candidates for counterexamples could be bilateral weighted shifts on in the spirit of those constructed by Bayart and Rusza in [7]. So we restate Question 1.1 as

###### Question 1.6.

Does there exist a frequently hypercyclic operator (necessarily living on some non-reflexive Banach space) which does not admit any invariant measure with full support?

Regarding Question 1.2, we are able to answer it in the negative. Let us first note that this question was already investigated in [3], where it was proved that there exists a bounded operator on the sequence space (actually, a weighted backward shift) which is frequently hypercyclic but does not admit any ergodic Gaussian measure with full support. The proof given in [3] can be easily extended to show that this frequently hypercyclic operator does not admit any ergodic measure with full support such that ; but the existence of a second order moment seems to be crucial in the argument. Our second main results shows that this aditional assumption can be dispensed with.

###### Theorem 1.7.

There exists a frequently hypercyclic operator on the space which does not admit any ergodic measure with full support.

However, Question 1.2 remains widely open on all other “classical” Banach spaces, and in particular for Hilbert space operators. In this special case there are several reasons to believe that the answer could be “Yes”, some of which will be outlined in the sequel. More generally, Theorem 1.5 makes the following instance of Question 1.2 especially interesting.

###### Question 1.8.

Does any frequently hypercyclic operator acting on a reflexive Banach space admit an ergodic probability measure with full support?

### 1.3 A natural parameter

One of the main ingredients in the proof of Theorem 1.7 is a certain parameter associated with any hypercyclic operator acting on a Banach space . Very roughly speaking, is the maximal frequency with which the orbit of a hypercyclic vector for can visit a ball centered at . In fact, for any we have

 c(T)=sup{c≥0;¯¯¯¯¯¯¯¯¯¯¯dens(NT(x,B(0,α))≥c for comeager manyx∈HC(T)}.

The parameter can bear witness of the existence of an ergodic measure with full support. Indeed, we can prove

###### Theorem 1.9.

Let be a linear dynamical system. If admits an ergodic measure with full support, then .

Theorem 1.7 follows from Theorem 1.9 combined with a result from [7] which (although not stated in this form) yields the existence of frequently hypercyclic bilateral weighted shifts on such that .

As it turns out, the parameter is also closely connected to the existence of distributionally irregular vectors for the operator . A vector is said to be distributionally irregular for if there exist two sets of integers , both having upper density , such that

 ∥Tix∥−−−→i→∞i∈A0and∥Tix∥−−−→i→∞i∈B∞.

This notion was studied by Bernardes, Bonilla, Müller and Peris in [9], where it is shown that the existence of a distributionally irregular vector for a linear system is equivalent to this system being distributionally chaotic in the sense of Schweitzer and Smítal [24].

In [9], the authors ask (a restricted form of) the following question. Assume that is a bounded operator on a complex Banach space and that has a perfectly spanning set of unimodular eigenvectors. Is it then true that admits a distributionally irregular vector? Since any such operator admits an ergodic measure with full support, the following result (whose proof makes use of the parameter in a crucial way) answers this question in the affirmative.

###### Theorem 1.10.

Let be a linear dynamical system, and assume that admits an ergodic measure with full support. Then admits a comeager set of distributionally irregular vectors. This holds in particular if is a complex Banach space and has a perfectly spanning set of unimodular eigenvectors.

The parameter already appears implicitely in the proof of Theorem 1.3 above, and it will be clear from this proof that its exact value is not easy to determine. In view of Theorem 1.9, the following question is quite natural. Note that a positive answer to this question, together with the result from [7] mentioned above, would answer Question 1.6 affirmatively.

###### Question 1.11.

Let be a linear dynamical system. Is it true that if is hypercyclic and admits an invariant measure with full support, then ?

The following special case is worth stating separately.

###### Question 1.12.

Let be a frequently hypercyclic operator acting on a reflexive Banach space . Is necessarily equal to ?

It is tempting to conjecture that the answer to this question is positive. In any event, since any operator  admitting an ergodic measure with full support satisfies by Theorem 1.9, this is in some sense a “necessary first step” towards a possibly affirmative answer to Question 1.8.

### 1.4 Baire category arguments

The following simple observation will be used throughout the paper: to obtain an ergodic measure with full support for an operator , it is in fact enough to find an invariant measure for such that . Indeed, the ergodic decomposition theorem then yields the existence of an ergodic measure such that . Being -invariant, this measure necessarily has full support because for each nonempty open set .

One of the most tempting roads one could follow in order to find an invariant measure such that is to try to use Baire Category arguments in the space of -invariant measures . This is far from being a new idea. Indeed, this strategy has already proved to be quite successful in a number of nonlinear situations; see for instance [25], [26] or [12].

Recall that for any Polish space , the space of all Borel probability measures on is endowed with the so-called Prokhorov topology, which is the weak topology generated by the space of all real-valued, bounded continuous functions on . In other words, a sequence of elements of converges to in if and only if for every . The topology of is Polish because is Polish (see e.g. [10, Chapter 2]). The set of all -invariant Borel probability measures on is easily seen to be closed in , and hence it is a Polish space in its own right.

For any Borel set , we denote by the set of of all Borel probability measures on which are supported on (i.e. such that ) and we set . If is open, then is easily seen to be in ; so is in .

Let us say that a Borel set is backward -invariant if . One can write as a countable intersection of backward -invariant open sets (just set , where is a countable basis of nonempty open sets for ) so that . Hence, a positive answer to the next question would solve Question 1.8 affirmatively.

###### Question 1.13.

Let be a frequently hypercyclic operator on a reflexive Banach space . Is it true that for every backward -invariant, nonempty open , the set is dense in ? Equivalently, is dense in ?

The two formulations of the question are indeed equivalent, because every backward -invariant open contains .

This very same question may be considered for chaotic operators. Recall that an operator on is said to be chaotic if it is hypercyclic and its periodic points are dense in . One of the most exciting open problems in linear dynamics is to determine whether every chaotic operator is frequently hypercyclic. This is widely open even in the Hilbert space setting, and closely related to an older question of Flytzanis [14] asking whether a hypercyclic operator on a Hilbert space whose unimodular eigenvectors of span a dense linear subspace of necessarily has uncountably many unimodular eigenvalues (or even a perfectly spanning set of unimodular eigenvectors). A positive answer to the next question would imply that in fact, every chaotic operator admits an ergodic measure with full support.

###### Question 1.14.

Let be a chaotic operator on a Banach space . Is it true that is dense in for any backward -invariant open set ? Equivalently, is dense in ?

It is worthing pointing out that the corresponding statement is known to fail in the nonlinear setting. Indeed, an example is given in [27] of a compact dynamical system with invertible, such that is topologically transitive with a dense set of periodic points, but admits no ergodic measure with full support. What makes this example especially interesting is that the map is in fact not frequently hypercyclic. This leads naturally to the following intriguing question.

###### Question 1.15.

Let be a compact dynamical system, and assume that is frequently hypercyclic. Does it follow that admits an ergodic measure with full support?

Our last result, Theorem 1.16 below, shows in particular that a weak form of Question 1.14 does have a positive answer for a large class of chaotic operators which, to the best of our knowledge, contains all known concrete chaotic operators. Note however that Theorem 1.16 cannot be of any use for showing that every chaotic operator has an ergodic measure with full support, since the assumption made therein that the operator  has a perfectly spanning set of unimodular eigenvectors already implies the existence of such a measure.

Let us say that a measure is a periodic measure for if it has the form

 ν=1NN−1∑n=0δTna,

where and satisfy . We will denote by the convex hull of the set of all periodic measures for . Equivalently, is the the set of all -invariant, finitely supported measures (which explains the notation). The closure of in is denoted by , and for any Borel set we set

Another family of invariant measures will be of interest for us. We shall say that a measure is a Steinhaus measure for if is the distribution of a random variable defined on some standard probability space , of the form

 Φ(ω)=∑j∈Jχj(ω)xj,

where the are unimodular eigenvectors for and is a finite sequence of independent Steinhaus variables (i.e. random variables uniformly distributed on the circle ). Any Steinhaus measure for is -invariant, by the rotational invariance of the Steinhaus variables. We denote by the family of all Steinhaus measures for , and by the closure of in . Accordingly, we set for any Borel set .

###### Theorem 1.16.

Let be a bounded operator on a complex separable Banach space .

• Assume that has a perfectly spanning set of unimodular eigenvectors, and that the periodic eigenvectors of are dense in the set of all unimodular eigenvectors. Then is a dense subset of .

• Assume “only” that has a perfectly spanning set of unimodular eigenvectors. Then is a dense subset of .

As explained above, the existence of an invariant measure supported on implies (and in fact, is equivalent to) the existence of an ergodic measure with full support. Hence, it follows in particular from part (b) that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support. This result is of course weaker than the one obtained in [5] since the ergodic measure has no reason for being Gaussian; but the the proof is quite different (being based on the Baire category theorem) and it looks much simpler than the existing ones from [16] and [5].

### 1.5 Organization of the paper

Section 1 is purely “nonlinear”. We first prove Theorem 1.5, and then we add a few simple remarks. In particular, it is shown in Proposition 2.15 that under rather mild assumptions, the existence of an invariant measure with full support implies the existence of a continuous measure with these properties. Theorem 1.3 is proved in Section 3. In the same section (Proposition 3.1), we also show that if an operator admits an invariant measure with full support, then the continuous, -invariant measures with full support form a dense subset of . The parameter is introduced in Section 4. This allows us to prove Theorems 1.7, 1.9 and 1.10 quite easily, together with two simple additional results: any frequently hypercyclic operator has a comeager set of “distributionally null” orbits; and the set of all frequently hypercyclic vectors for a given operator is always meager in the underlying Banach space (this was obtained independently in [7] and [21]). The proof of Theorem 1.16 is given in Section 5. It makes use of another result, Theorem 5.2, which provides several necessary and sufficient conditions for the existence of an invariant measure supported on and belonging to the closure of a family of invariant measures satisfying some natural assumptions. Using Theorem 5.2, we also prove two additional results similar to Theorem 1.16 which give some plausibility to the conjecture that every chaotic operator is frequently hypercyclic and in fact admits an ergodic measure with full support. We conclude the paper by listing several equivalent formulations of the perfect spanning property.

## 2 Construction of invariant measures with full support

In this section, we prove Theorem 1.5. As already mentioned in the introduction, it is a well-known fact that frequently hypercyclic continuous self-maps of a compact metric space admit invariant measures with full support. Since the proof of Theorem 1.5 uses in a crucial way the idea of the proof in the compact case, we first sketch the latter briefly. We refer to [15, Lemma 3.17] for more details.

### 2.1 The compact case

Let be a frequently hypercyclic continuous self-map of a compact metrizable space , and let . Denote by the space of real-valued continuous functions on . By the Riesz representation theorem, one can identify with the set of all positive linear functionals on such that , where denotes the function constantly equal to . The latter is -compact as a subset of , and it is also metrizable because is separable. For each , let be defined as

 μN:=1NN∑n=1δTnx0,

where, for each , is the Dirac mass at . Since all are probability measures, one can find an increasing sequence of integers and a probability measure such that tends to in the -topology of as tends to infinity; in other words,

 1NkNk∑n=1f(Tnx0)⟶∫Xfdm for every f∈C(X).

Since for any , we see that

 ∫X(f∘T)dμNk−∫XfdμNk⟶0 for every f∈C(X).

It follows that for every , so the measure is -invariant.

Let be a nonempty open set in , and choose a nonempty open set whose closure is contained in . Since is closed in , the map is upper semi-continuous on ; so we have

 m(¯¯¯¯V)≥limsupk→∞μNk(¯¯¯¯V).

Since , and recalling that , it follows that

 m(U)≥liminfN→∞1N#{n∈[1,N] ; Tnx0∈V}>0.

This shows that the measure has full support.

It is clear that compactness is crucial in the above proof, since essentially everything relies on the Riesz representation theorem. The metrizability of is also needed for two reasons: it implies that is separable, so that we can extract from the sequence a -convergent sequence (but this is merely a matter of convenience); and it allows to identify the linear functionals on with the Borel measures on .

### 2.2 Proof of Theorem 1.5

In the proof of Theorem 1.5, we will use the same idea as above to associate with each -compact set a Borel probability measure on . Then the measure will be obtained as the supremum of all these measures . In what follows, we denote by the family of all -compact subsets of . We also fix a frequently hypercyclic point for .

Since we will have to consider simultaneously all sets , a diagonalization procedure will be needed. To avoid extracting infinitely many sequences of integers, it is convenient to consider a suitable invariant mean on , the space of all bounded sequences of real numbers. Recall that an invariant mean is a positive linear functional on such that and for every and all , where is the translated sequence .

It is not hard to see that there exists an invariant mean such that

 m(ϕ)≥liminfn→∞1nn∑i=1ϕ(i) for all ϕ∈ℓ∞(N).

For example, one may take , where is a non-principal ultrafilter on . In the sequel, we fix such an invariant mean . In order to emphasize the fact that should be viewed as a finitely additive measure on , we write the result of the action of on a “function” as an integral:

 m(ϕ)=∫Nϕ(i)dm(i).

 ∫Nϕ(i+a)dm(i)=∫Nϕ(i)dm(i) for every a∈N.

Before really starting the proof of Theorem 1.5, let us observe that the topologies and have the same Borel sets. Indeed, since each point has a neighbourhood basis consisting of -compact sets and since the topology is Lindelöf (being separable and metrizable), every -open set is a countable unions of -compact sets and hence is -Borel. So it makes sense to speak of Borel measures on without referring explicitly to one of the topologies or .

We now start the proof of Theorem 1.5 with the following fact, that will allow us to deal with Borel measures when applying the Riesz representation theorem.

###### Fact 2.1.

Every -compact subset of is -metrizable.

###### Proof.

Let us fix , and let be a countable basis of (nonempty) open sets for with respect to the topology . For each , the -closure of is -compact. For each pair with , one can find a -continuous function such that on and on . It is clear that the (countable) family of all such functions separates the points of ; and since is -compact, metrizability follows.∎

For each , let us denote by the space of all -continuous, real-valued functions on . Using the same argument as in Section 2.1 above, we can now prove

###### Fact 2.2.

For every , there exists a unique positive Borel measure on such that

The measure satisfies . Moreover, if has nonempty interior with respect to the topology , then .

###### Proof.

The first part is obvious by the Riesz representation theorem, since the formula

 L(f)=∫N(1Kf)(Tix0)dm(i),f∈C(K,τ)

defines a positive linear functional on . It is also clear that the measure  thus defined satisfies . Now, let us denote by the -interior of in , and assume that . Then

 μK(K) = ∫N1K(Tix0)dm(i)≥liminfn→∞1nn∑i=11K(Tix0)≥liminfn→∞1nn∑i=11V(Tix0)>0

because is frequently hypercyclic for . ∎

For each , we extend the measure to a positive Borel measure on (still denoted by ) in the usual way; that is, we set for every Borel set . Then .

The following simple yet crucial fact will allow us to define a measure on as the supremum of all measures .

###### Fact 2.3.

If and if , then .

###### Proof.

Since is Polish, every Borel measure on is regular. So it is enough to show that for every -compact set . We do this in fact for every -compact set . (Recall that the topology is finer than , so every -compact set is -compact).

Since is -compact, we may replace with , i.e we may assume that . Since is -compact, the function (the indicator function of ) is upper-semicontinuous with respect to the topology , when considered as a function on . So, by the metrizability of , one can find a decreasing sequence of functions of such that converges to pointwise on . Of course, the restrictions of the functions to belong to . Since and we have

 ∫KfjdμK = ∫N(1Kfj)(Tix0)dm(i)≤∫N(1Lfj)(Tix0)dm(i)=∫KfjdμL

for all . Letting tend to infinity on both sides, we get . ∎

From Fact 2.3 and since the family of -compact sets is closed under finite unions, we see that the family is what is sometimes called “filtering increasing”. From this, we can easily deduce

###### Fact 2.4.

If we set

 μ(A):=supK∈KτμK(A) for % every Borel set A⊂X,

then is a positive Borel measure on , such that .

###### Proof.

Obviously for every Borel set and . It is also clear that for every increasing sequence of Borel sets . So we just have to check that is finitely additive.

Let be Borel subsets of with . Since for all , we get that by the very definition of . Conversely, we have by Fact 3

 μK(A)+μK′(A′)≤μK∪K′(A)+μK∪K′(A′)=μK∪K′(A∪A′)≤μ(A∪A′)

for any , and hence . ∎

We now check that has the required properties.

###### Fact 2.5.

The measure is -invariant and has full support.

###### Proof.

The fact that has full support is obvious by Fact 2.2: if is a nonempty -open subset of , then contains a -compact set with nonempty -interior and hence .

The main point is to show that is -invariant. For this, it is in fact enough to show that for every -compact set . Suppose indeed that it is the case. Then, by the regularity of the Borel measures and , we have for every Borel set . Applying this inequality to now yields that , and since this means that . So we get for every Borel set , which proves that is -invariant.

Let us fix . We are first going to prove that

 μK(T−1(E))≤μT(K)(E) for every K∈Kτ.

Note that this makes sense because, as is continuous with respect to the topology , belongs to . Since is closed in , there exists a decreasing sequence of functions in such that converges to pointwise on . Then we have

 μT(K)(E) = ∫T(K)1E∩T(K)dμT(K)=limj→∞∫T(K)fjdμT(K) = limj→∞∫N(1T(K)fj)(Tix0)dm(i)=limj→∞∫N(1T(K)fj)(Ti+1x0)dm(i),

the last equality being true because is an invariant mean on . Now observe that since is nonnegative on , we have

 (1T(K)fj)(Ti+1x0)≥(1K⋅(fj∘T))(Tix0).

So we get, using the fact that the function is -continuous on ,

 ∫N(1T(K)fj)(Ti+1x0)dm(i) ≥ ∫N(1K⋅(fj∘T))(Tix0)dm(i)=∫K(fj∘T)dμK.

Therefore, we obtain

 μT(K)(E) ≥ limj→∞∫K(fj∘T)dμK=∫K(1E∩T(K)∘T)dμK.

Since , this yields that

 μT(K)(E) ≥ ∫K1T−1(E)∩KdμK=μK(T−1(E)),

which proves our claim. Since , it follows that for every , and hence that . This concludes the proof of Fact 2.5. ∎

If we normalize the measure  by setting , we have thus proved that is a -invariant Borel probability measure on with full support, and this completes the proof of Theorem 1.5. ∎

###### Remark 2.6.

What we have in fact proved is the following result. Let be a Polish dynamical system and let be any topology on which is coarser that the original topology but with the same Borel sets, whose compact sets are metrizable, and such that is continuous with respect to . Then, for any invariant mean on and any point , one can find a -invariant finite Borel measure on such that for every -compact set .

###### Remark 2.7.

The measure constructed above may not be a probability measure, so we do have to normalize it. Indeed, we have

 μ(X)=supK∈KτμK(K)=supK∈Kτ∫N1K(Tix0)dm(i)

and this may very well be smaller than if

### 2.3 There are many invariant measures with full support

It is worth mentioning that as soon as there exists at least one -invariant measure with full support, the set of all such measures is in fact a large subset of in the Baire Category sense. This (well-known) observation will be needed below.

###### Lemma 2.8.

Let be a Polish dynamical system, and denote by the set of all -invariant Borel probability measures on with full support. If , then is a dense subset of .

###### Proof.

Let be a countable basis of (nonempty) open sets for . Then a measure has full support if and only if for all ; and since the maps are lower semi-continuous on , it follows that is in . Now, assume that , and let us choose any element of . If is arbitrary, then the measure is -invariant for any , and it has full support because has full support; in other words, . Since as , this shows that is dense in . ∎

### 2.4 Ergodic measures with full support

In view of Questions 1.8 and 1.15, it is of course natural to wonder whether the above construction can give rise to an ergodic  measure with full support. There is no reason at all that the measure  constructed in the proof of Theorem 1.5 should be ergodic. Still, if we were able to prove directly that , then we would get for free an ergodic probability measure with full support for : indeed, as already explained in the introduction, it would follow directly from the ergodic decomposition theorem that there exists an ergodic probability measure  such that (in fact by ergodicity), and such a measure  would necessarily have full support. However, we see no reason either that the measure  constructed in the proof of Theorem 1.5 should satisfy . The next proposition clarifies this a little bit.

###### Proposition 2.9.

Let be a Polish dynamical system. Assume that is endowed with a Hausdorff topology coarser than the original topology such that every point of has a neighbourhood basis (with respect to the original topology) consisting of -compact sets, and that is a homeomorphism of . Then, the following assertions are equivalent.

1. admits an ergodic measure with full support;

2. there exists a point such that

 limN→∞\rm dens––––––––––N⋃r=0(NT(x0,V)−r)=1for\;every\;open\;set\;V≠∅.

Moreover, if a point satisfies (2), then the measure constructed in the proof of Theorem 1.5 starting from satisfies .

###### Proof.

Assume that admits an ergodic measure with full support. By the pointwise ergodic theorem and since the space is second-countable, -almost every point satisfies

 liminfn→∞1nn∑i=11U(Tix0)≥ν(U)foreveryopensetU≠∅.

Let us fix such a point . Since the sum in the left-hand side is equal to the cardinality of the set , we have for every open set . Applying this with for a given open set and , and observing that for every , it follows that

 \rm dens––––––––––N⋃r=0(NT(x0,V)−r)≥ν(N⋃r=0T−r(V))foreveryopensetV≠∅.

Since by ergodicity, this shows that (2) is satisfied.

To conclude the proof, it is now enough to show that if a point satisfies (2), then the measure constructed in the proof of Theorem 1.5 starting from satisfies .

With the notation of the proof of Theorem 1.5, we have for any and every integer , because is assumed to be a homeomorphism of . So we may write

 μ(N⋃r=0T−r(K)) ≥ μ⋃Nr=0T−r(K)(N⋃r=0T−r(K)) ≥ liminfn→∞1nn∑i=11⋃Nr=0T−r(K)(Tix0) = dens–––––{i∈N ; Tix0∈N