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Chaotic systems are characterized by sensitive dependence on initial conditions, dense trajectories and dense periodic points. Due to their occurrence in a wide variety of dynamical systems, they can usually be rigorously studied only if accompanied by some assumptions on the geometric / differential properties of the transformation. One geometric structure which is well understood is the class of Anosov diffeomorphisms or more generally, completely hyperbolic systems and to some extent, partially hyperbolic systems. A weaker version of these rather stringent assumptions is the concept of invariant cones. This is an invariant structure in the tangent bundle of a map and we will explore various consequences of this invariant structure. We will mostly focus on a broad class of maps on the torus and derive various properties and also discuss the interplay between them, under the assumption that a "dominated invariant cone system" is present. |