Oscillating Statistics of Transitive Dynamics

We prove that topologically generic orbits of C ⁰ , transitive and non-uniquely ergodic dynamical systems, exhibit an extremely oscillating asymptotical statistics. Precisely, the minimum weak* compact set of invariant probabilities that describes the asymptotical statistics of each orbit of a residual set contains all the ergodic probabilities. If besides f is ergodic with respect to the Lebesgue measure, then also Lebesgue-almost all the orbits exhibit that kind of extremely oscillating statistics.

• Measure Preserving Maps
• Dynamical Systems
• Ergodic Theory
• Asymptotic Statistics

1. Introduction

We will study the statistical average for typical orbits of transitive dynamics, under a non traditional viewpoint.

On the one hand, the traditional viewpoint studies the limit in the future of the Birkhoff averages, starting always from the same initial point, and for Lebesgue-positive sets of orbits in the future. So, under this tradi- tional viewpoint, the “statistics” of the system (at least for C²-dynamical systems with some kind of hyperbo- licity), is mainly obtained from the existence of physical measures, of Sinai-Ruelle-Bowen (SRB) measures, and of Gibbs measures (see for instance the survey [3] ).

Relevant advances on the study of the asymptotic behavior of the time-averages from the traditional view- point can be found for instance in the following articles. In [8] Viana and Yang study the existence of physical measures for partially hyperbolic systems with one-dimensional center direction. Bonatti’s survey [2] gives an overview of the state of art in the theme of the asymptotical dynamics of C¹-differentiable systems from the topological viewpoint. In [5] Liverani proves that piecewise C² expanding maps may exhibit Gibbs measures without needing the bounded distortion property.

On the other hand, instead of adopting the traditional viewpoint, along this paper we will study the time averages that start at any future iterate of the initial point. This viewpoint is based on a philosophical argument: the way that the observers in the future will perceive the forward statistics of the system, is not the way that it is computed today. In fact, today the observers compute the Birkhoff average along the finite future piece orbit of length n (which we like to call “the clima”), by the mean value of the observable functions from time 0 to n. But the observers in the future―who will live, say, at time―will compute their Birkhoff average along the finite piece of orbit of length n (i.e. they will perceive their clima), by the mean value of the observable functions between time m and time.

This non-traditional viewpoint of studying the Birkhoff averages and their limits (i.e. the statistics) does not give preferences to different initial observation instants. So, our conclusions include also the prediction of all the climas that the observers in the future will perceive.

The key result is Theorem 2:

Topologically typically, the clima observed at infinitely many times in the future must widely differ from the clima observed at present time, provided that the dynamics is deterministic (non hazardous), transitive and non- niquely ergodic.

This is an unexpected result, taking into account that the system is autonomous and deterministic. Nevertheless, the idea of the proof of Theorem 2 is extremely simple. The route of its proof is the result of join- ing the following three simple observations. First, if the system is transitive, then its topologically generic orbits in the future are dense. Second, for any ergodic measure, and for any -typical point, the Birkhoff average starting at converges to. So, for any, for any fixed n sufficiently large, and for any point x close enough, the Birkhoff average starting at x is -near. Third, any dense orbit in the future has such an iterate x close enough.

Thus, one concludes that the Birkhoff averages, with fixed n but starting at different points in the future of the same orbit, oscillate among all the ergodic measures of f, when.

Even if the main theorem is the consequence of the latter simple observations, and no more proof than the above argument would be needed, we will include all the details of this proof (see Section 3) to be readable by a wide class of scientists and students.

1.1. Mathematical Background

Let M be a compact manifold of finite dimension. Let be continuous. We consider the dynamical system obtained by iteration of f in the future, i.e. the family of orbits with initial condition. This dynamical system is composed by the solutions of the recurrent equation.

We denote by the space of all the probability measures in M, endowed with the weak^* topology (see for instance Definition 6.1 of [7] ). That is, if is a sequence of probability measures in M, we define

where is the space of continuous real functions in M, with the supremum norm.

Recall that a measure is invariant by f if for any Borel-measurable set. We denote by the space of f-invariant probability measures, and by the set of ergodic probability measures for f.

We recall that if and only if

(See for instance Theorem 6.8 of [7] ).

To each initial state, or equivalently to each orbit, we associate the double-indexed sequence of non necessarily invariant probability measures, which we call empirical probabilities, defined by:

where is the Dirac-Delta probability measure supported on the point. In other words, the empirical probability is the probability distribution that is observed during a statistical experiment on which one computes the Birkhoff average (i.e. the temporal average) of the observable functions along a finite piece of the orbit of x, from time m to time. Precisely:

We agree to call the double-indexed sequence of empirical probabilities the complete future statistics of the orbit of x. For the sake of concision we call it the statistics of x.

Since the space is metrizable and weak^*-compact, it is sequentially compact (see for instance Theorems 6.4 and 6.5 of [7] ). Thus, any sequence of empirical probabilities has convergent subsequences when.

We agree to call the set of all limit probability measures of all such sequences of empirical probabilities in, the asymptotical statistics of the orbit of x (Definition 2.1).

1.2. Statement of the Results

If preserves the Lebesgue measure m and is ergodic, then the sequence is conver-

gent for Lebesgue-almost all (see for instance Theorem 6.12 (ii) of [7] ). In other words, its limit set is a singleton. Also, if there exists a unique physical measure whose basin of statistical attraction covers Lebesgue almost all the points, or if there exists a unique SRB-like measure, then the limit set of the sequence

is a singleton for Lebesgue-almost all (see [4] ).

In contrast, if instead of restricting to the case, we consider all the sequences of the form where, then the limit set may be non convergent, and moreover, extremely oscillating (see Definition 2.3). In fact, in this paper we prove the following result:

Theorem 1. Let be continuous, preserving the Lebesgue measure of M and ergodic with respect to it, but non uniquely ergodic. Then Lebesgue-almost all the orbits of f have extremely oscillating asymptotical statistics. Precisely, it contains all the ergodic probability measures of f.

Let us state a similar result that holds for maps that do not preserve the Lebesgue measure. In Theorem 3.6 of

[1] , Abdenur and Andersson studied the limit set of the sequence for Lebesgue-almost all the

orbits of C⁰-generic maps. Such generic systems do not preserve the Lebesgue measure. They proved that the

particular sequence of empirical probabilities is convergent for Lebesgue-almost all. So, its limit set is a singleton.

Now, for transitive and non-uniquely systems, we observe all the sequences

instead of restricting to the case. Let us apply a topological criterium instead of a Lebesgue- probabilistic criterium when selecting the relevant orbits of the system. With such an agreement, we say that an orbit is generic if it belongs to a residual set in M. Then the asymptotical statistics is far from being a singleton: it is extremely oscillating. In fact, we prove the following result:

Theorem 2. Let be continuous, transitive and non uniquely ergodic. Then generic orbits of f have extremely oscillating asymptotical statistics. Precisely, any ergodic probability for f belongs to the asymp- totical statistics of each generic orbit.

Theorems 1 and 2 imply the necessary extremely changeable “clima”, i.e. the time averages of the observable functions along finite pieces of all the relevant orbits in the ambient manifold M vary so much in the long term, to approach all the extremal invariant probabilities of the system (the ergodic measures). Even if the system is fully deterministic and it is governed by an autonomous and unchangeable recurrence equation, even if the parameters in this equation are fixed, even if the states along the deterministic orbit are not perturbed, no topologically relevant orbit of the system has a predictable statistics along its long-term future evolution. On the contrary, its asymptotical statistics is extremely changeable in the long-term future, exhibiting at least, as many probability distributions as ergodic measures of f exist.

This paper is organized as follows: In Section 2 we state the precise mathematical definitions to which the results refer, and in Section 3 we include the proofs of Theorems 1 and 2.

2. Definitions

Since the double-indexed sequence of empirical probabilities completely describes de statis- tics (i.e. the time-average) of any finite piece of the orbit of x, the limit set in the space of probabilities describes what we call the asymptotical statistics of the orbit, according to the following definition:

Definition 2.1. (Asymptotical statistics in the space of probabilities)

The asymptotical statistics of the orbit of, which we denote by, is the set composed by all the limits in of the convergent subsequences of any sequence of empirical probabilities of x, where is any mapping from the set of natural numbers to itself. Precisely:

Following the classical Krylov-Bogolioubov construction of invariant probabilities (see for instance the proofs of Theorems 6.9, 6.10, and Corollary 6.9.1 of [7] ), it is standard to check that:

In other words, the asymptotical statistics of x is a nonempty compact set of probability measures which are invariant by f.

Definition 2.2. (Convergent or oscillating asymptotical statistics)

The orbit is statistically convergent if its asymptotical statistics is composed by a unique proba- bility measure, i.e.

It is statistically oscillating if it is non convergent.

We recall that f is called uniquely ergodic if (see for instance [6] ).

Definition 2.3. (Extremely oscillating asymptotical statistics)

When f is non-uniquely ergodic we say that the orbit is statistically extremely oscillating if its asymptotical statistics contains all the f-invariant ergodic probability measures. Namely:

Definition 2.4. (Transitive system) The dynamical system by iterates of is called transitive if for any pair of nonempty open sets in M there exists a positive iterate of U that intersects V.

Let us denote to the topology of M, i.e. the family of all the open sets of M. So, is transitive, by definition, if

where denotes the set of positive integer numbers. Equivalently,

Recall that M is a finite dimensional manifold. So, is transitive if and only if there exists whose orbit in the future is dense in M.

Definition 2.5. (Residual sets and generic orbits)

According to Baire-category theory a set is said residual if it contains a countable intersection of open and dense subsets of M. It is standard to check that the countable intersection of residual sets is residual. Since M is a compact manifold, any residual set R is dense, but not all dense sets are residual.

Given a residual set we say that the orbits are generic.

3. The Proofs

The weak^* topology of the space of probability measures is metrizable (see for instance Theorem 6.4 of [7] ). We choose and fix a weak^*-metric in, which we denote by dist.

To prove Theorems 1 and 2 we first state the following lemmas:

Lemma 3.1. Let be continuous and denote the space of f-invariant probability measures. Let. Let and let be the asymptotical statistics of the orbit of x, according to Definition 2.1. Then

Proof: From equality (4), if and only if:

for some sequence such that. This condition holds if and only if for any there exists such that

Since, for any there exists such that for all. We deduce that

In other words, if and only if for all and all, the point x belongs to the set

From equality (2) note that

Then

We have proved that, if and only if for all and all, the point x belongs to the set

We conclude that, where the set is defined by equality (5), ending the proof. ,

Lemma 3.2. If is ergodic, then for all there exists such that

Proof: We take any continuous real function. By Birkhoff Ergodic Theorem and from the definition of ergodicity (see [7] ), we have

From equality (3) we obtain

The last equality holds for all. So, by the condition (1) which defines the weak^* topology in the space of probability measures, we deduce:

Therefore, for, for all there exists such that

We conclude that

Now, it is left to prove that, for fixed, fixed, and fixed, the set is open in the ambient manifold M. Since is open in the space of probability measures, it is enough to check that the mapping:

is continuous. So, let us prove that for any convergent sequence, the image sequence converges to in the weak^* topology, where

To apply condition (1) we consider any continuous real function. From equality (3)

Since is continuous and we have

We deduce that

From condition (1) we conclude the following equality in the space of probability measures:

showing that the mapping is continuous, and ending the proof of Lemma 3.2. ,

To prove Theorem 1, we first state the following:

Lemma 3.3. Let be continuous, preserve the Lebesgue measure m, be ergodic with respect to m and be non-uniquely ergodic. If is an ergodic probability measure for f, then the set defined by equality (5) of Lemma (3.1) has total Lebesgue measure. Thus, for Lebesgue almost all.

Proof: Denote by m the Lebesgue measure of the manifold M, after a rescaling to make.

From Lemma 3.2, for all there exists such that

Thus

Define

We conclude that is a nonempty open set in M. By construction. Since the Lebesgue measure m is ergodic, we deduce that

But the set is nonempty and open, and the Lebesgue measure is positive on nonempty open sets. So

So, taking we deduce that

Note that if then, and for any there exists such that. Thus

But the converse inclusion is obvious because for all, we obtain particular values of. Thus

In brief, we have proved that

Substituting by its expression in equality (6) we conclude:

Finally, applying Lemma 3.1 we conclude

as wanted. ,

End of the proof of Theorem 1.

Proof: Fix with. Since the space of probability measures is weak^*-compact, the closure is compact. So, there exist a finite covering

of with open balls of radius and centered in. Since the radius is fixed, it is not restrictive to take for all.

Since is ergodic for f, we can apply Lemma 3.3, to deduce that the following set

has total Lebesgue measure.

Take any. Since covers there exists such that. Therefore, by the triangle inequality:

We deduce that

Since has full Lebesgue measure, then the set

also has full Lebesgue measure. From (9) we have

We deduce that for Lebesgue-almost all, the following assertion holds for any ergodic measure:

Since for any there exists such that, we have

Applying Lemma 3.1, we deduce that. We have proved that for Lebesgue-almost all any ergodic measure belongs to. After Definition 2.3, the asymptotical statistics of the orbit of x is extremely oscillating. We conclude that Lebesgue-almost all the orbits exhibit extremely oscillating asympto- tical statistics, as wanted.,

Now, to prove Theorem 2, we state the following:

Lemma 3.4. If is continuous and transitive and if is an ergodic invariant measure for f, then the set defined in Lemma 3.1 is residual. Thus, for generic.

Proof:

From Lemma 3.2, for all the set

Consider the set defined by equality (6):

Since f is continuous and transitive, from Definition 2.4 we obtain that, for any nonempty open set, there exists such that. In other words, is dense in M. So is open and dense.

So, taking and applying Definition 2.5, we deduce that

Applying again Definition 2.5:

For all there exists such that, and thus

We deduce that

is also residual in M. In other words

Substituting by its expression in equality (6) we conclude:

Finally, applying Lemma 3.1 we conclude as wanted. ,

End of the proof of Theorem 2.

Proof: Fix with and construct the finite covering of by equality (7), and the set defined by equality (8). Since the measures are ergodic for f, we can apply Lemma 3.4, to deduce that the set is residual for all.

Take any. Since covers there exists such that. Therefore, by the triangle inequality, we deduce assertion (9).

Since is residual, then the set is also residual. From (9) we have

We deduce that for generic, the following assertion holds for any ergodic measure:

Since for any there exists such that, we have

Applying Lemma 3.1, we deduce that. We have proved that for generic any ergodic measure belongs to. After Definition 2.3, the asymptotical statistics of the orbit of x is extremely oscillating. We conclude that the generic orbits of f exhibit extremely oscillating asymptotical statistics, as wanted. ,

Acknowledgements

The author thanks the Editor and the anonymous Referee. She thanks the partial support of “Agencia Nacional de Investigación e Innovación” (ANII), “Comisión Sectorial de Investigación Cientfica” (CSIC) of “Universidad de la República”, and “Premio L’Oréal-UNESCO-DICYT” (the three institutions of Uruguay).

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