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In this paper we generalize the method of Lagrangian descriptors to two dimensional, area preserving, autonomous and nonautonomous discrete time dynamical systems. We consider four generic model problems-a hyperbolic saddle point for a linear, area-preserving autonomous map, a hyperbolic saddle point for a nonlinear, areapreserving autonomous map, a hyperbolic saddle point for linear, area-preserving nonautonomous map, and a hyperbolic saddle point for nonlinear, area-preserving nonautonomous map. The discrete time setting allows us to evaluate the expression for the Lagrangian descriptors explicitly for a certain class of norms. This enables us to provide a rigorous setting for the notion that the 'singular sets" of the Lagrangian descriptors correspond to the stable and unstable manifolds of hyperbolic invariant sets, as well as to understand how this depends upon the particular norms that are used. Finally we analyze, from the computational point of view, the performance of this tool for general nonlinear maps, by computing the "chaotic saddle" for autonomous and nonautonomous versions of the Hénon map.

The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems Keywords: Topological entropy, Li-Yorke chaos, Lyapunov exponent, continua, nonautonomous systems, difference equations, ordinary and partial differential equations, set-valued dynamical system, global attractor, non-autonomous and random pullback attractors.

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A spaceXis said to be almost totally disconnected if the set of its degenerate components is dense inX. We prove that an almost totally disconnected compact metric space admits a minimal map if and only if either it is a finite set or it has no isolated point. As a consequence we obtain a characterization of minimal sets on dendrites and local dendrites.

This article is devoted to the study of invariant ε-scrambled sets. We show that every topologically mixing map with at least one fixed point contains at least one such set. Additionally we show that this condition can be weakened in the case of symbolic dynamics, e.g. mixing can be replaced by transitivity. Some relations between mixing and proximal relation are also studied.