Exotic Attractors

1 Attractors in Dynamical Systems.- 1.1 Introduction.- 1.2 Basic definitions.- 1.3 Topological and dynamical consequences.- 1.4 Attractors.- 1.5 Examples and counterexamples.- 1.6 Historical remarks and further comments.- 2 Liapunov Stability and Adding Machines.- 2.1 Introduction.- 2.2 Adding Machines and Denjoy maps.- 2.3 Stable Cantor sets are Adding Machines.- 2.4 Adding Machines and periodic points: interval maps.- 2.5 Interlude: Adding Machines as inverse limits.- 2.6 Stable ?-limit sets are Adding Machines.- 2.7 Classification of Adding Machines.- 2.8 Existence of Stable Adding Machines.- 2.9 Historical remarks and further comments.- 3 From Attractor to Chaotic Saddle: a journey through transverse instability.- 3.1 Introduction.- 3.1.1 Riddled and locally riddled basins.- 3.1.2 Symmetry and invariant submanifolds.- 3.2 Normal Liapunov exponents and stability indices.- 3.2.1 Normal Liapunov exponents.- 3.2.2 The normal Liapunov spectrum.- 3.2.3 Stability indices.- 3.2.4 Chaotic saddles.- 3.2.5 Strong SBR measures.- 3.2.6 Classification by normal spectrum.- 3.3 Normal parameters and normal stability.- 3.3.1 Parameter dependence of the normal spectrum.- 3.3.2 Global behaviour near bifurcations.- 3.4 Example: ?2-symmetric maps on ?2.- 3.4.1 The spectrum of normal Liapunov exponents.- 3.4.2 Global transverse stability for f.- 3.4.3 Global transverse stability for g.- 3.5 Example: synchronization of coupled systems.- 3.5.1 Electronic experiments.- 3.5.2 Observations.- 3.5.3 Analysis of the dynamics.- 3.6 Historical remarks and further comments.