We introduce a new definition of distinguished trajectory that generalizes the concepts of fixed point and periodic orbit to aperiodic dynamical systems. This new definition is valid for identifying distinguished trajectories with hyperbolic and nonhyperbolic types of stability. The definition is implemented numerically and the procedure consists of determining a path of limit coordinates. It has been successfully applied to known examples of distinguished trajectories. In the context of highly aperiodic realistic flows our definition characterizes distinguished trajectories in finite time intervals, and states that outside these intervals trajectories are no longer distinguished. This paper attempts to generalize the concepts of fixed point and periodic orbit to time dependent aperiodic dynamical systems. Fixed points and periodic orbits are keystones for describing solutions of autonomous and time periodic dynamical systems, as the stable and unstable manifolds of these hyperbolic objects form the basis of the geometrical template organizing the description of the dynamical system. The mathematical theory of aperiodic dynamical systems is far from complete. In this context, this work deals with a general definition that encompasses the concepts of fixed point and periodic orbit and which when applied to finite time and aperiodic dynamical systems identifies special trajectories that play an organizing role in the geometry of the flow.

We discuss dark energy models which might describe effectively the actual acceleration of the universe. More precisely, for a 4-dimensional Friedmann-Lemaître-Robertson-Walker (FLRW) universe we consider two situations: First of them, we model dark energy by phantom energy described by a perfect fluid satisfying the equation of state P = (β − 1)ρ (with β < 0 and constant). In this case the universe reaches a "Big Rip" independently of the spatial geometry of the FLRW universe. In the second situation, the dark energy is described by a phantom (generalized) Chaplygin gas which violates the dominant energy condition. Contrary to the previous case, for this material content a FLRW universe would never reach a "big rip" singularity (indeed, the geometry is asymptotically de Sitter). We also show how this dark energy model can be described in terms of scalar fields, corresponding to a minimally coupled scalar field, a Born-Infeld scalar field and a generalized Born-Infeld scalar field. Finally, we introduce a phenomenologically viable model where dark energy is described by a phantom generalized Chaplygin gas. PACS numbers: 98.80.-k,98.80.Es,11.10.-z PU-ICG-04/09, astro-ph/0404540 * Electronic address: mariam.bouhmadi@port.ac.uk † Electronic address: madrid@iaa.es 1 We would like to mention that there are other candidates for dark energy based on brane-world models [15] and modified 4dimensional Einstein-Hilbert actions[16], where a late time acceleration of the universe may be achieved.

In this paper we study the regularity properties of fractional maximal operators acting on BVfunctions. We establish new bounds for the derivative of the fractional maximal function, both in the continuous and in the discrete settings.Question A. (Haj lasz and Onninen [10]) Is the operator f → |∇M f | bounded from W 1,1 (R d ) to L 1 (R d )?A standard dilation argument reveals the true nature of this question: whether the variation of the maximal function is controlled by the variation of the original function, i.e. if we haveProgress on this problem has been restricted to dimension d = 1. For the uncentered maximal operator (defined similarly as in (1.1), with the supremum taken over all balls containing the point x in its closure), which we denote here by M , Tanaka [22] showed that if f ∈ W 1,1 (R) then M f is weakly differentiable andIn the case 1 ≤ β < d, Question B admits a positive answer, which follows from the main result of Kinnunen and Saksman in their aforementioned work [13]. In fact, [13, Theorem 3.1] states the following regularizing effect: if f ∈ L r (R d ) with 1 < r < d and 1 ≤ β < d/r, then M β f is weakly differentiable and C(d, β) is a universal constant. In our case, given 1 ≤ β < d and f ∈ W 1,1 (R d ), by the Sobolev embedding we have f ∈ L p * (R d ), where p * = d/ (d − 1), and hence f ∈ L r (R d ) for any 1 ≤ r ≤ p * . We may choose r with 1 < r < d such that 1 ≤ β < d/r and hence (1.5) holds.ThenQuestion B (in the case 0 ≤ β < 1) is the main motivation for this work. The presence of the fractional part introduces additional difficulties as we shall see in the course of the paper (e.g. one does not necessarily have M β (f )(x) ≥ |f (x)| a.e.). Here we give a positive answer to Question B in dimension d = 1 for the uncentered