Semi-Hyperbolicity and Bi-Shadowing

Preface In the beginning of 90s the authors of this monograph proposed a generalization of the concept of hyperbolicity, first for differentiable mappings and later for Lipschitz mappings, which they called 'semi-hyperbolicity'. This arose indirectly in the context of their research at that time on the effect of spatial discretization on the behavior of a dynamical system, in particular that of finite machine arithmetic in a computer representation of a dynamical system , and rapidly broadened into a series of papers in which differing aspects and applications of the concept were explored. These papers form the basis of this monograph, the aim of which is to present a more thorough and systematic development of the concept of semi-hyperbolicity, as well as to illustrate its practicality. While the connection with the theory of hyperbolic systems is important and will not be neglected, much of the motivation of the authors comes from their interest and background in applications of dynamical systems and this has naturally influenced the types of questions asked and investigated. As it often happens, our everyday duties and new interests at long last distracted us from semi-hyperbolicity, bi-shadowing and all such things. So, the manuscript, almost completed, remained not finished formally. To our surprise and pleasure, the whole theme of semi-hyperbolicity and bi-shadowing did not die. There appears a number of brilliant investigations. So, we decided to finalize, at least formally, the manuscript and make it available on the Web. Real world dynamical processes can be extremely complicated, yet numerous quite simple idealized mathematical models often appear to capture the essence of what is happening and allow useful predictions to be made. This fact, which is to some extent justifiable mathematically, has been central to the resounding success of rationalist scientific thinking over the past four hundred years. Moreover, a large amount of mathematical analysis owes its origin to the development of concepts and tools needed to formulate and investigate such idealized mathematical models, which, though relatively simple, are by no means trivial. Traditionally mathematical models have involved differential equations, both ordinary and partial, thus when relevant representing continuous time dynamical systems on appropriately chosen state spaces. In contrast, much of the modern theory of dynamical systems has focussed on discrete time dynamical systems generated by iteration of a mapping f of a given state space X into itself, that is on difference equations Though seemingly less complicated, their …