For an ergodic measure preserving dynamical system $(X, \cal B, \mu, T)$ and a measurable function $f$ with values in $\mathbb{Z}$, we consider for $x \in X$ the set of values of the ergodic sums $S_nf(x):= \sum_0^{n-1} f(T^k x), n \geq 1$.

If $f$ is integrable with $\mu(f) > 0$, several properties of this set (from the point of view of recurrence or arithmetic sets) are simple consequences of Bourgain’s results (1989).

For example, the set ${S_nf(x), n \geq 1}$ contains infinitely many squares for a.e. $x$. If $f$ is not integrable, this property may fail, as shown by a construction of M. Boshernitzan. We give also a counter-example of an integrable centered function $f$ for which the cocycle $(S_nf(x), n \geq 1)$ is non regular and the property fails.

Time reversal symmetry arises in many dynamical systems. In particular, it is an important aspect of dynamical systems which emerge from physical theories such as classical mechanics, thermodynamics and quantum mechanics. In this talk, we introduce the notion of a reversible dynamical system in symbolic dynamics. We investigate conjugacy invariants of a topological Markov chain which possesses an involutory reversing symmetry.