_{1}

^{*}

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.

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 C^{2}-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 [

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 [^{1}-differentiable systems from the topological viewpoint. In [^{2} 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

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

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.

Let M be a compact manifold of finite dimension. Let

We denote by ^{*} topology (see for instance Definition 6.1 of [

where

Recall that a measure

We recall that

(See for instance Theorem 6.8 of [

To each initial state

where

We agree to call the double-indexed sequence

Since the space ^{*}-compact, it is sequentially compact (see for instance Theorems 6.4 and 6.5 of [

We agree to call the set of all limit probability measures of all such sequences of empirical probabilities in

If

gent for Lebesgue-almost all

In contrast, if instead of restricting to the case

Theorem 1. Let

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

[

orbits of C^{0}-generic maps. Such generic systems do not preserve the Lebesgue measure. They proved that the

particular sequence

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

instead of restricting to the case

Theorem 2. Let

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.

Since the double-indexed sequence of empirical probabilities

Definition 2.1. (Asymptotical statistics

The asymptotical statistics of the orbit of

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 [

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

It is statistically oscillating if it is non convergent.

We recall that f is called uniquely ergodic if

Definition 2.3. (Extremely oscillating asymptotical statistics)

When f is non-uniquely ergodic we say that the orbit

Definition 2.4. (Transitive system) The dynamical system by iterates of

Let us denote

where

Recall that M is a finite dimensional manifold. So,

Definition 2.5. (Residual sets and generic orbits)

According to Baire-category theory a set

Given a residual set

The weak^{*} topology of the space ^{*}-metric in

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

Lemma 3.1. Let

Proof: From equality (4),

for some sequence

Since

In other words,

From equality (2) note that

Then

We have proved that,

We conclude that

Lemma 3.2. If

Proof: We take any continuous real function

From equality (3) we obtain

The last equality holds for all^{*} topology in the space

Therefore, for

We conclude that

Now, it is left to prove that, for fixed

is continuous. So, let us prove that for any convergent sequence^{*} topology, where

To apply condition (1) we consider any continuous real function

Since

We deduce that

From condition (1) we conclude the following equality in the space

showing that the mapping

To prove Theorem 1, we first state the following:

Lemma 3.3. Let

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

From Lemma 3.2, for all

Thus

Define

We conclude that

But the set

So, taking

Note that if

But the converse inclusion is obvious because for all

In brief, we have proved that

Substituting

Finally, applying Lemma 3.1 we conclude

as wanted. ,

End of the proof of Theorem 1.

Proof: Fix ^{*}-compact, the closure

of

Since

has total Lebesgue measure.

Take any

We deduce that

Since

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

We deduce that for Lebesgue-almost all

Since for any

Applying Lemma 3.1, we deduce that

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

Lemma 3.4. If

Proof:

From Lemma 3.2, for all

Consider the set

Since f is continuous and transitive, from Definition 2.4 we obtain that, for any nonempty open set

So, taking

Applying again Definition 2.5:

For all

We deduce that

is also residual in M. In other words

Substituting

Finally, applying Lemma 3.1 we conclude

End of the proof of Theorem 2.

Proof: Fix

Take any

Since

We deduce that for generic

Since for any

Applying Lemma 3.1, we deduce that

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).

EleonoraCatsigeras, (2015) Oscillating Statistics of Transitive Dynamics. Advances in Pure Mathematics,05,534-543. doi: 10.4236/apm.2015.59049