Abundance of wild historic behavior, generalized physical measures

# Abundance of wild historic behavior, ergodic decomposition and generalized physical measures

V. Araujo and V. Pinheiro V. Araujo, Departamento de Matemática, Universidade Federal da Bahia
V. Pinheiro, Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil.
July 14, 2019
###### Abstract.

Using Caratheodory measures, we associate to each positive orbit of a measurable map , a Borel measure . We show that is -invariant whenever is continuous or is a probability. We use these measures to prove the Ergodic Decomposition Theorem and also to characterize the generic points of an invariant measure. These measures are used to study the historic points of the system, that is, points with no Birkhoff averages, and we construct topologically generic subset of wild historic points for wide classes of dynamical models. Using these tools, we extend the notion of physical measure and show that the generic homeomorphism has such a generalized physical measure. We use properties of the measure to deduce some features of the dynamical system involved, like the existence of heteroclinic connections from the existence of open sets of historic points.

###### Key words and phrases:
historic behavior, physical measure, generalized physical measure, ergodic decomposition, generic homeomorphism, heteroclinic attractor
###### 1991 Mathematics Subject Classification:
Primary: 37A99, 37C20. Secondary: 37C29
Work carried out at the Federal University of Bahia, and IMPA. Partially supported by CNPq, PROCAD/CAPES, IMPA and PRONEX - Dynamical Systems

## 1. Introduction

The asymptotic behavior of a dynamical system given by some transformation is in general quite complex. To understand the behavior of the orbit of a point we should focus on the -limit set , the set of accumulation points of . The number of iterates of the positive orbit of near some given point depends on and, in fact, the limit frequency of visits of the orbit of to some subset of the phase space is non-existent in general!

A simple and very general example is obtained taking a non-periodic point , an strictly increasing integer sequence and the subset

 A={fn1(x),…,fn2(x)}∪{fn3(x),…,fn4(x)}∪⋯=∪i≥0{fn2i+1(x),…,f2(i+1)(x)}.

Then it is straightforward to check that

Thus even if exists, we still have that does not exist, In Ergodic Theory behavior of this type is bypassed through the use of weak convergence in Birkhoff’s Ergodic Theorem, ensuring that exists in the weak topology for almost every point with respect to any -invariant measure.

As defined by Ruelle in [50] and Takens in [56] we say that has “historic behavior” if the sequence does not converge in the weak topology.

The above construction of non-convergent sequence of frequency of visits can be easily obtained in any topological Markov chain and so, through Markov partitions and their coding, can also be obtained in every hyperbolic basic set of an Axiom A diffeomorphism; see e.g. Bowen and Ruelle in [17].

As shown by Takens [56] (and also Dowker [21]) the set of points with historic behavior is residual. Hence, generically the points of a basic piece of an Axiom A diffeomorphism has both dense orbit and historic behavior! More recently, Liang-Sun-Tian [35] extend this result for the support of non-uniformly hyperbolic measures for smooth diffeomorphisms and Kiriki-Ki-Soma [32] obtain a residual historic subset in the basin of attractor of any geometric Lorenz model. Here we extend and strengthen these results to whole classes of dynamical models.

We say that a measure which gives positive mass to the subset of points with historic behavior is a historic measure.

The construction of the heteroclinic attractor attributed to Bowen, as presented in [55], as a flow with an open subset of points with historic behavior, provides an example of such phenomenon where Lebesgue measure is a non-invariant historic measure; see Example 2 in Section 3. Other similar examples can be found in Gaunersdorfer [22]. In the quadratic family the following result from Hofbauer-Keller provides many other examples.

###### Theorem 1.1 ([28]).

For the family , there exists a non-denumerable subset of parameters in such that Lebesgue measure is an historic measure with respect to for all .

We emphasize that Jordan, Naudot and Young showed in [29], through a classical result from Hardy [26], that if time averages of a bounded observable do not converge, then all higher order averages do not exist either, that is, every Césaro or Hölder higher order means fail to converge; see e.g. [27] for the definitions of these higher order summation processes. This shows that historic points cannot be regularized by taking higher order averages.

This paper aims at studying the ergodic properties of historic measures. For that we consider a set function that will serve as a premeasure to obtain a Borel measure through a classic well-known construction of Carathéodory; a thorough presentation of which can be found in Rogers [49]. This is a way to bypass failure of convergence of time averages in certain classes of systems, in particular in the “Bowen eye” example.

The following Block and Lyubich result provides a natural notion of ergodicity for non-invariant measures.

###### Theorem 1.2 ([14]).

Lebesgue measure is ergodic (but non-invariant) for every -unimodal transformation (in particular for the quadratic family) which has no periodic attractor.

Using this notion we obtain: for an ergodic measure (not necessarily invariant) for -almost every . In addition, for the quadratic family there exists a -invariant measure such that for Lebesgue almost every .

We also reprove the Ergodic Decomposition Theorem (see e.g. Mañé [36]) using as ergodic components with respect to every -invariant probability measure.

These measures can be used to define an extension of the notion of physical measure, and using a recent result from Abdenur and Andersson [1] we show that these generalized physical measures do exist for the generic homeomorphism of any compact boundaryless manifold.

We show that if is purely atomic for an open subset of points , then the system has similar dynamics to the heteroclinic Bowen attractor; see [55, 22]. In particular Lebesgue measure is historic.

Many more results about the set of points with historic behavior are known. To the best of our knowledge, Pesin and Pitskel were the first to show that these points carry full topological pressure and satisfy a variational principle for full shits, in [46]. These points are named “non-typical” in [12] by Barreira-Schmeling who show that this set has full Hausdorff dimension and full topological entropy for subshifts of finite type, conformal repellers and conformal horseshoes. These points are said “irregular points” in the work [57] of Thompson where it is shown that, for maps with the specification property having some point with historic behavior, the set of such irregular points carries full topological pressure. This indicates that this set of points is not dynamically irrelevant. Genericity of these “irregular points” for dynamics with specification and non-convergence of time averages for an open and dense family of continuous functions follows from [20, Proposition 21.18]; see Lemma 5.4 in Subsection 5.1.3 and also [9, 7, 8, 6].

The Hausdorff dimension of subsets of such points is studied by Barreira-Saussol [11] for hyperbolic sets for smooth transformations and also by Olsen-Winter in [43] in the setting of multifractal analysis for subshitfs of finite type and for several specific transformations also by Olsen [41, 40] and generalized by Zhou-Chen [59].

Recently Kiriki and Soma [33] show that every surface diffeomorphism exhibiting a generic homoclinic tangency is accumulated by diffeomorphisms which have non-trivial wandering domains whose forward orbits have historic behavior. More recently Laboriau and Rodrigues [34] motivated by the ideas in [33] present an example of a parametrized family of flows admitting a dense subsets of parameter values for which the set of initial conditions with historic behaviour contains an open set. These are examples of persistent historic behavior.

In this paper we also provide broad classes of examples where has infinite mass on every open subset. These points with wild historic behavior are also topologically generic for Axiom A systems, for expanding measures, topological Markov chains, suspension semiflows over these maps, Lorenz-like attractors and many other dynamical models. These are the “points with maximal oscillation” studied by Denker, Grillenberger and Sigmund in [20, Proposition 21.18]; see Lemma 5.7 in Subsection 5.1.4. Olsen in [42, 39] studied “extreme non-normal” numbers and continued fractions which are, in particular, wild historic points, and so form a topologically generic subset; see Example 3 in Section 5.1.

We note that there are classes of systems with no specification where our construction can be performed: see Example 5 and Remark 5.12 in Subsection 5.7. Hence, the genericity of wild historic points is more general than the genericity of points with maximal oscillation in systems with specification.

## 2. Statement of results

### 2.1. Measure associated to a sequence of outer measures

Let be a compact metric space, a measurable map with respect to the Borel -algebra and let be the distance on . Let be a sequence of finitely additive outer probabilities on , so that each is a set function defined on all parts of with values in and such that .

###### Definition 1.

Given a sequence of finitely additive outer measures with unit total mass, we define

 τ(A)=limsupn→∞1nn−1∑j=0ξj(A).

It is easy to see that

 τ(A)≤τ(A∪B)≤τ(A)+τ(B)

for all and consequently

 τ(n⋃j=1Aj)≤n∑j=1τ(Aj), (1)

for all finite collection of subsets of .

However, one can easily find examples of countable collections of sets and of measures such that

 τ(⋃j∈NAj)>∑j∈Nτ(Aj).
###### Example 1.

If the positive -orbit is infinite then, taking and the Dirac point mass at , i.e. , we get for each and so .

Let be the collection of open sets of . According to the definition of pre-measure of Rogers [49], restricted to is a pre-measure; see [49, Definition 5].

###### Definition 2.

Given , define

 ν(Y)=supr>0νr(Y)(=limr↘0νr(Y)),

where and is the set of all countable covers of by elements of with .

The function , defined on the class of all subset of , is called in [49] the (Caratheodory) metric measure constructed from the pre-measured by Method II (Theorem 15 of [49]).

We define to be the restriction of to the Borel sets, i.e., . From [49, Theorem 23] (see also [49, Theorem 3]) it follows that is a countable additive measure defined on the -algebra of Borel sets.

Certain measure-theoretical properties of and those of are provided in Section 3: first with no dynamical assumptions, and then assuming that is -invariant.

### 2.2. Ergodic decomposition with measures generated by an orbit

Considering the particular choice with for a given , it is easy to see that the associated set function satisfies for every subset of . We denote by the measure obtained from with the above choices.

Using the properties of versus we prove the following version of the Ergodic Decomposition Theorem in Section 4.2.

###### Theorem A.

Let be a perfect compact metric space and let be a measurable map. If is an -invariant finite measure then, for almost every , is a -invariant ergodic probability. Furthermore, That is,

for every (in particular, for every Borel set ).

This is a similar to [36, Theorem 6.2] but our proof is different and our approach provides a natural generalization of the notion of physical measure.

### 2.3. Wild historic behavior generically

###### Example 2.

We consider the well-known example of a planar flow with divergent time averages attributed to Bowen; see Figure 1 and [55]. We set the time- map of the flow, and define with for , where is a point in the interior of the plane curve formed by the heteroclinic orbits connecting the fixed hyperbolic saddle points .

We assume that this cycle is attracting, that is, the homoclinic connections are also the set of accumulation points of the positive orbit of ; in particular, we have near the stable/unstable manifold of the saddles . However, this accumulation is highly unbalanced statistically.

Indeed, if we denote the expanding and contracting eigenvalues of the linearized vector field at by and and at by and , and the modulus associated to the upper and lower saddle connection by

 λ=α−/β+andσ=β−/α+

then , and , since the cycle is assumed to be attracting; see [55], where it is proved that for every continuous function with we have

 limsupn→+∞1nn−1∑j=0φ(xj) =σ1+σφ(A)+11+σφ(B),and liminfn→+∞1nn−1∑j=0φ(xj) =λ1+λφ(B)+11+λφ(A).

In particular, for any small

 limsupn→+∞1nn−1∑j=0χBε(A)(xj)≥σ1+σandlimsupn→+∞1nn−1∑j=0χBε(B)(xj)≥λ1+λ,

so that and for all . Since we can find arbitrarily small such that are in , then we deduce

 2>ηx0(X)=ηx0(A)+ηx0(B)=σ1+σ+λ1+λ=σ+λ+2λσ1+σ+λ+λσ>1

because .

Moreover we also obtain for Lebesgue almost every point , so that there is no Physical/SRB measure.

###### Definition 3.

We say that for which is not a probability measure is a historic point or a point with historic behavior. We denote the set of historic points of the map by .

This definition follows the one in Takens [56]. The existence of such points can be obtained in any compact invariant set of a map which is conjugate to a full shift, e.g. an horseshoe. It can be easily adapted to any topological Markov chain (Markov subshift of finite type) and hence can be applied to any basic set of an Axiom A diffeomorphism; see e.g. [53, 17, 16] for the definitions of Axiom A diffeomorphisms and [56].

In fact, Dowker [21] showed that if a transitive homeomorphism of a compact set admits a point with transitive orbit and historic behavior, then historic points form a topologically generic subset of . As observed in [56], the generic subset of points with historic behavior exists also in the stable set of any basic set of an Axiom A diffeomorphism.

Combining the construction in [56] with density of hyperbolic points we can obtain a stronger property for a topologically generic subset points.

###### Definition 4.

We say that for which gives infinite mass to every open subset of a compact invariant subset for the dynamics, is a point with wild historic behavior in or a wild historic point. We denote the set of wild historic points of the map by .

The following result provides plenty of classes of examples of abundance of wild historic behavior.

###### Theorem B.

The set of points with wild historic behavior in

1. every mixing topological Markov chain with a denumerable set of symbols (either one-sided or two-sided);

2. every open continuous transitive and positively expansive map of a compact metric space;

3. each local homeomorphism defined on an open dense subset of a compact space admitting an induced full branch Markov map;

4. suspension semiflows, with bounded roof functions, over the local homeomorphisms of the previous item;

5. the stable set of any basic set of either an Axiom A diffeomorphism, or a Axiom A vector field;

6. the support of an expanding measure for a local diffeomorphism away from a non-flat critical/singular set on a compact manifold;

7. the support of a non-atomic hyperbolic measure for a diffeomorphism, or a vector field, of a compact manifold;

is a topologically generic subset (denumerable intersection of open and dense subsets).

We stress that item (2) provides mild conditions to obtain a generic subset of wild historic points: positively expansive maps are those for which there exists so that any pair of distinct points are guaranteed to be -apart in some finite time, that is, there exists so that (also know as “sensitive dependence”). This property together with the existence of a dense orbit in a compact metric space ensures the existence of plenty wild historic points.

For detailed definitions of positively expansive, induced full branch Markov map, expanding measure, Axiom A systems and hyperbolic measures, see Section 5 and references therein.

The connection between properties of real numbers in the interval and properties of the orbits of the maps for any integer and the Gauss map , together with the natural coding of the dynamics by topological Markov chains, enables us to easily relate wild historic behavior with absolutely abnormal numbers and extremely non-normal continued fractions; see e.g. [40, 41, 43] and references therein. Theorem B contains some of the genericty results in these works as particular cases; see e.g. Example 3 in Section 5.1.

We have obtained a topologically generic subset of wild historic points for abundant classes of systems with (non uniform) hyperbolic behavior. This indicates that the absence of wild historic points implies that all invariant probability measures are either atomic or non-hyperbolic, that is, no wild historic behavior forces every non-atomic invariant probability measure to have zero Lyapunov exponents, either for diffeomorphisms, vector fields or endomorphisms of compact manifolds; and even for local diffeomorphims away from a singular set. The only extra assumptions being sufficient smoothness (Hölder- seems to be enough) and a singular/critical set regular enough. We present some conjectures in Subsection 2.6.

### 2.4. Generalized physical measures

Let be a measurable map defined on a compact metrical space . Given a reference measure , we may ask if there is an invariant measure such that

 limn1n+∞∑j=0φ∘fj(x)=∫φdν (2)

for every continuous function and every point in some set with .

More generally, we may ask if there is a set with and an invariant measure so that

 ∫U(limn1n+∞∑j=0φ∘fj(x))dμ=∫φdν (3)

for every continuous function .

If is a Riemann compact manifold and the reference measure is the Lebesgue measure, i.e. , a measure satisfying (2) above is called a physical measure. Moreover, if (or is absolutely continuous with respect to the Lebesgue measure on the unstable leaves), is called a SRB measure.

###### Definition 5.

We say than is a -physical measure if (2) holds for a set with positive measure.

###### Definition 6.

We say than is a generalized -physical measure if (3) holds for a set with .

As in the invariant case, a non invariant measure is called ergodic if or for every invariant set (i.e., . There are many examples of non invariant ergodic measure. For instance, Martens [37] proved that the Lebesgue measure is ergodic for every -unimodal map without periodic attractor. More generally, it was shown in [47] that every (non invariant) expanding measure with bounded distortion can be decomposed into a finite number of (non invariant) ergodic measures.

The following provides a necessary and sufficient condition for existence of generalized -physical measures.

###### Theorem C.

Let be continuous. Then admits a generalized -physical measure if, and only if, . If, in addition, is ergodic (but not necessarily invariant), then admits a -physical measure.

We prove this in Section 6.1 after studying the behavior of the measure in the setting of continuous dynamics, done in Section 6. In Section 6.2 we obtain necessary and sufficient conditions for the existence of a -physical measure.

The notion of generalized physical measure has some resemblances with the notions of “Iliashenko’s statistical attractors”, “Milnor’s attractors” and “SRB-like measures”; see [19, 18] and references therein. However, our definition is not restricted to using Lebesgue measure as a reference measure and our construction is significantly different.

#### 2.4.1. Generalized physical measures exist for the generic homeomorphism

Recently Abdenur and Andersson [1] show that

###### Theorem 2.1.

For the generic homeomorphism of a compact boundaryless finite manifold Birkhoff time averages

 limn→+∞1nn−1∑j=0φ(fj(x))

exist Lebesgue almost everywhere for every continuous function .

From the properties of and obtained for continuous dynamics in Section 6, we see that Theorem 2.1 ensures that, for a generic subset of homeomorphisms , the measure is a -invariant probability measure for Lebesgue almost every . Hence we deduce

###### Corollary D.

The generic homeomorphism of a compact boundaryless finite manifold admits generalized Lebesgue-physical measures.

This shows that our definition of generalized physical measure is relevant for a large family of continuous dynamics.

In Example 6 of Section 6 we provide an example of a smooth systems with no Lebesgue-generalized physical measure.

### 2.5. Strong historic behavior and heteroclinic attractors

The following result shows that strong historic behavior in the neighborhood of two hyperbolic periodic points is sufficient to obtain an heteroclinic connection relating the two points.

###### Theorem E.

Let be a diffeomorphism on a compact boundaryless manifold endowed with a pair of hyperbolic periodic points satisfying, for some

• is atomic with two atoms for every either in or .

Then and have a heteroclinic cycle: and .

That is, this result gives a sufficient condition for a diffeomorphism to exhibit an heteroclinic attractor, as in the example of Bowen as presented in [55] and Example 2.

### 2.6. Conjectures

We believe that it is possible to use properties of the measures to understand certain dynamical features of the system involved.

As a simple example, we observe that for a continuous map of a compact space, implies for all (e.g., an irrational circle rotation or torus translation).

Indeed, given , every point admits a neighborhood such that and so for every . We obtain an open cover of the compact and so a finite subcover exists. Hence .

Another observation is that if for all , then for a diffeomorphism of a compact surface or a endomorphism of the circle or interval.

Indeed, in those cases, ensures the existence of a horseshoe for ; see [30]. In both cases, these invariant subsets are conjugated to a full shift with finitely many symbols, and so if is always a finite measure we contradict Theorem B.

It is then natural to pose the following

###### Conjecture 1.

Every smooth () diffeomorphism of a compact manifold satisfying for all has zero topological entropy.

Note that the examples provided by Beguin, Crovisier and Le Roux in [13] show that this conjecture is false if we allow the dynamics to be just a homeomorphism.

It is known in many cases that points with historic behavior form a geometrically big subset (full Hausdorff dimension) of the dynamics; see e.g. Barreira-Schmeling [12]. Recently Zhou and Chen [60] show that the set of historic points carries full topological pressure for systems with non-uniform specification under certain conditions. Here we have shown that wild historic points are generic in a broad class of examples.

###### Conjecture 2.

In the class of examples considered in Theorem B the set of wild historic points has full Hausdorff dimension, full topological entropy and full topological pressure.

Developing our observation about absence of wild historic behavior after Theorem B, we propose the following.

###### Conjecture 3.

Absence of wild historic points for a smooth enough () dynamics of a diffeomorphism or a vector field in a compact manifold implies that all invariant probability measures are either atomic or have only zero Lyapunov exponents.

An analogous conclusion holds for all smooth enough () local diffeomorphisms away from a critical/singular set which is sufficiently regular (non-flat).

Note that from Peixoto’s Theorem [45, 44, 24, 25] for an open and dense subset of vector fields in the topology on compact orientable surfaces, for all , the limit set of every orbit is contained in one of finitely many hyperbolic critical elements (fixed points or periodic orbits). Hence wild historic points are absent from an open and dense subset of smooth continuous time dynamics on surfaces. Thus, for vector fields Conjecture 3 makes sense only on manifolds of dimension or higher.

We propose the following weakening of Condition (H) from Theorem E.

###### Conjecture 4.

For a diffeomorphism of a compact manifold of dimension or higher, , if there exists a point and two hyperbolic periodic points and so that

 ηx=a⋅1π(P)π(P)∑j=1δfjP+b⋅1π(P)π(Q)∑j=1δfjQ,

where give the minimal periods of , then is accumulated in the topology by diffeomorphisms so that the continuations of the periodic points are homoclinically related.

### 2.7. Organization of the text

We present the construction of the measure through a pre-measure in a very general dynamical setting in Section 3. Then we specify the construction of the measure for a given orbit of a point in Section 4, where we also deduce Theorem A on ergodic decomposition of invariant measures using such frequency measures . We construct the generic subset of wild historic points in Section 5 proving Theorem B and providing abundant classes of examples to apply our results. In Section 6, we consider continuous transformations and relate our constructions with the notion of physical measure and extend this to the notion of generalized physical measure, proving Theorem C and Corollary D. Finally, in Section 7 we prove Theorem E.

### Acknowledgments

Both authors were partially supported by CNPq (V.A. by projects 477152/2012-0 and 301392/2015-3; V.P. by project 304897/2015-9), FAPESB and PRONEX-Dynamical Systems (Brazil). V.P. would also like to thank the finantial support from Balzan Research Project of J. Palis.

## 3. The measure η versus the premeasure τ

Here we provide some properties of the premeasure and of the measure without assuming invariance of .

###### Lemma 3.1.

If is a uncountable collection of pairwise disjoint compact sets of , then there is a countable sub-collection such that for every .

###### Proof.

Let be the family of such that , for , and let . If is uncountable, then there is some such that is uncountable. As is an increasing function, it follows that for all and all . Set and let be the collection of such that for each .

As is uncountable, there are infinitely many such that is uncountable. Thus, let be such that is uncountable and, for each , let be the collection of such that for each .

Let be any finite collection of elements of with for . Let also be such that for . For we have

 1≥1ss−1∑i=0ξs(L)=1s4n−1∑t=0s−1∑i=0ξi(Bε(Kt))>4n12n=2

a contradiction. Hence is countable and, as for every the proof is complete. ∎

###### Lemma 3.2.

If is a compact set, then .

###### Proof.

First suppose that . Given let be an open cover of such that . As is compact there is a finite subcover of . Of course, . Let be such that . As is finite, we get . As a consequence, .

Now, suppose that . Let be a finite cover of . Given , let be such that . Let . Note that and . Thus, and, as a consequence, . ∎

###### Corollary 3.3.

If is a uncountable collection of pairwise disjoint compact subsets of , then there is a countable sub-collection such that for every .

###### Lemma 3.4.

If is an open set with then .

###### Proof.

Let be a positive number. As is regular, let be small so that , where . By lemma 3.2, we can take so that .

Let be small enough so that . Let be such that . Thus, , or equivalently,

 ∣∣∣η(A)−∑V∈Cτ(V)∣∣∣≤δ/2.

As and , changing by if necessary, we may assume that .

Now since is compact, there is some finite open cover of of . Therefore, and . As is finite, we get and so,

 τ(A′)≤η(A)+δ/2.

As , we get . As a consequence,

 τ(¯¯¯¯A)≤η(A)+δ

for every . ∎

In what follows we define to be the subfamily of open subsets such that ; and then define

 G={U∈G0:η(∂f−1(U))=0}.
###### Lemma 3.5.

If is a perfect set (every point is an accumulating point), then generates the topology of (and also the Borel sets). Furthermore, if is a perfect set and is continuous, then generates the topology of .

###### Proof.

It is enough to show that for any given and any , with , there is a sequence such that .

We consider for any fixed . Then is a pairwise disjoint collection of compact subsets of with positive measure. From Corollary 3.3 this collection must be countable. Hence there exists such that . Since can be taken arbitrarily close to zero, this proves what we need.

For a continuous we replace by in the definition of and note that is compact since is open. The same argument above applies in this case and completes the proof. ∎

###### Lemma 3.6.

If is a probability, then for each .

###### Proof.

We assume that is a probability. As , it follows from lemma 3.4 that for every . Hence .

Now we assume, by contradiction, that there is some such that . As , we get , a contradiction, completing the proof. ∎

###### Corollary 3.7.

If is a basis of the topology, then

 ηx(X)=1⟺ηx(A)=τx(A),∀A∈U.
###### Corollary 3.8.

If is a probability, then for every closed set .

###### Proof.

Choose so that for each . Thus . ∎

Given as before and a bounded measurable function, we define

 τ(φ):=limsupn→+∞1nn−1∑j=0ξi(φ)=limsupn→+∞1nn−1∑j=0∫φdξj.

Clearly and so, for any finite collection of functions . We need the following result.

###### Lemma 3.9.

Let be a finite regular measure. If is a continuous function, then there exists a sequence of simple function such that , the are pairwise disjoint for each , for -almost all and also .

###### Proof.

See any standard textbook on measure and integration, e.g. [Rudin]. ∎

###### Lemma 3.10.

If is a probability, then for each continuous function .

In particular, we obtain that is a Borel probability measure.

###### Proof.

First we show that for every continuous function .

Indeed, if is a continuous function, let be given by Lemma 3.9. Let , and . Note that and are nonnegative functions and . Since we get . Thus, using that we have

 ∫ψndη =ℓn∑j=1(aj,n−a)η(Aj,n)=ℓn∑j=1(aj,n−a)τ(Aj,n) ≥τ(ℓn∑j=1(aj,n−a)χAj,n)=τ(ψn).

As implies that and since , we get

 ∫φdη−a=∫ψdη≥τ(ψ)=τ(φ)−a.

That is, for every continuous function .

Now, suppose that for some continuous function . As is also a continuous function . Therefore

 1 =∫1dη=∫((1−φ)+φ)dη=∫(1−φ)dη+∫φdη >τ(1−φ)+τ(φ)≥τ((1−φ)+φ)=τ(1)=1.

This contradiction completes the proof. ∎

We note that up to this point we have not used any invariance relation for the measures or .

### 3.1. The invariant case

Now we assume that for all Borel subsets of . This depends on the choice of the sequence and must be checked for each specific case.

###### Lemma 3.11.

If is a probability measure, then is -invariant.

###### Proof.

By assumption, we have both and on the Borel -algebra, thus is -invariant. ∎

###### Lemma 3.12.

Let be a compact set. Given there are a finite collection of disjoint open sets with and such that and .

###### Corollary 3.13.

The following properties are equivalent:

1. is a probability;

2. for each continuous function ;

3. exists for each continuous function .

###### Proof.

First we assume that is a probability and let . By Lemma 3.10 applied to we get

 ∫φdη =−(∫−φdη)=−(limsupn→∞1nn−1∑j=0ξj(−φ)) =−(−liminfn→∞1nn−1∑j=0ξj(φ))=liminfn→∞1nn−1∑j=0ξj(φ).

On the other hand, applying Lemma 3.10 to , we obtain

 ∫φdη=τ(φ)=limsupn→∞1nn−1∑j=0ξj(φ).

This is enough to see that item (2) is true if item (1) is true. Clearly (2) implies (3).

Now we assume that (3) is true. Suppose that . Let be small enough so that . Let be a finite collection of disjoint open sets of with diameters smaller than and such that . Note that