Oscillating statistics of transitive dynamics

We prove that topologically generic orbits of C0 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.


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 traditional viewpoint, the "statistics" of the system (at least for C 2 -dynamical systems with some kind of hyperbolicity), 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 viewpoint 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 C 1 -differentiable systems from the topological viewpoint. In [5] Liverani proves that piecewise C 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 0 m > -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 m n + . 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 nonniquely 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 joining 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 µ , and for any µ -typical point 0 x , the Birkhoff average starting at 0 x converges to µ . So, for any 0 >  , for any fixed n sufficiently large, and for any point x close enough 0 x , the Birkhoff average starting at x is  -near µ . Third, any dense orbit in the future has such an iterate x close enough 0 x . 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 n → +∞ .
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. 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]

Mathematical Background
where y δ is the Dirac-Delta probability measure supported on the point y M ∈ . In other words, the empirical probability

Statement of the Results
preserves the Lebesgue measure m and is ergodic, then the sequence ∈ (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 [4]).
In contrast, if instead of restricting to the case Let us apply a topological criterium instead of a Lebesgueprobabilistic 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: 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 asymptotical 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.

Definitions
Since the double-indexed sequence of empirical probabilities 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: is weak -compact, and . We recall that f is called uniquely ergodic if # 1 f =  (see for instance [6]).

Definition 2.3. (Extremely oscillating asymptotical statistics)
When f is non-uniquely ergodic we say that the orbit where +  denotes the set of positive integer numbers. Equivalently,

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: