Basic Topological Concepts of Dynamical Systems

  • Viacheslav Z. GrinesEmail author
  • Timur V. Medvedev
  • Olga V. Pochinka
Part of the Developments in Mathematics book series (DEVM, volume 46)


The theory of dynamical systems extensively uses a lot of concepts and tools from other branches of mathematics: topology, algebra, geometry etc. In this chapter we review the basic definitions and facts, necessary for understanding the presented results. We begin the fundamental notions of a set and of a map, we describe which structures one has to define on a set to make it a group, a linear space, a metric space. We recall the main properties of maps and their spaces. We give some facts on embedding of a surface into a 3-manifold and the definition of a wild embedding. We show how the universal cover is constructed and the connection between Nielsen-Thurston’s theory and structurally stable diffeomorphisms of surfaces.


Wild Embeddings Minimal Equivalence Relation Negative Euler Characteristic Nielsen Classes Moebius Strip 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Viacheslav Z. Grines
    • 1
    Email author
  • Timur V. Medvedev
    • 2
    • 3
  • Olga V. Pochinka
    • 1
  1. 1.Department of Fundamental MathematicsNational Research University Higher School of EconomicsNizhny NovgorodRussia
  2. 2.Department of Differential Equations, Mathematical and Numerical AnalysisNizhny Novgorod State UniversityNizhny NovgorodRussia
  3. 3.Laboratory of Algorithms and Technologies for Networks AnalysisNational Research University Higher School of EconomicsNizhny NovgorodRussia

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