Narrow Results

In this paper we consider a lattice dynamical system generated by a parabolic equation modeling suspension flows. We prove the existence of a global compact connected attractor for this system and the upper semicontinuity of this attractor with respect to finite-dimensional approximations. Also, we obtain a sequence of approximating discrete dynamical systems by the implementation of the implicit Euler method, proving the existence and the upper semicontinuous convergence of their global attractors.

In this paper, we study the mean return times to a given set for suspension flows. In the discrete time setting, this corresponds to the classical version of Kac’s lemma [11] that the mean of the first return time to a set with respect to the normalized probability measure is one. In the case of suspension flows we provide formulas to compute the mean return time. Positive measure sets on cross sections are also considered. In particular, this varies linearly with continuous reparametrizations of the flow and takes into account the mean escaping time from the original set. Relation with entropy and returns to positive measure sets on cross sections is also considered.