Topological Dynamical Systems

Preface
Notation
0 Introduction
	0.1 Definition and a (very brief) historical overview
	0.2 Continuous vs. discrete time
	0.3 The dynamical systems point of view
	0.4 Examples
1 Basic notions
	1.1 Invariant and periodic points
	1.2 Invariant sets
	1.3 Transitivity
	1.4 Limit sets
	1.5 Topological conjugacy and factor mappings
	1.6 Equicontinuity and weak mixing
	1.7 Miscellaneous examples
2 Dynamical systems on the real line
	2.1 Graphical iteration
	2.2 Existence of periodic orbits
	2.3 The truncated tent map
	2.4 The double of a mapping
	2.5 The Markov graph of a periodic orbit in an interval
	2.6 Transitivity of mappings of an interval
3 Limit behaviour
	3.1 Limit sets and attraction
	3.2 Stability
	3.3 Stability and attraction for periodic orbits
	3.4 Asymptotic stability in locally compact spaces
	3.5 The structure of (asymptotically) stable sets
4 Recurrent behaviour
	4.1 Recurrent points
	4.2 Almost periodic points and minimal orbit closures
	4.3 Non-wandering points
	4.4 Chain-recurrence
	4.5 Asymptotic stability and basic sets
5 Shift systems
	5.1 Notation and terminology
	5.2 The shift mapping
	5.3 Shift spaces
	5.4 Factor maps
	5.5 Subshifts and graphs
	5.6 Recurrence, almost periodicity and mixing
6 Symbolic representations
	6.1 Topological partitions
	6.2 Expansive systems
	6.3 Applications
7 Erratic behaviour
	7.1 Stability revisited
	7.2 Chaos(1): sensitive systems
	7.3 Chaos(2): scrambled sets
	7.4 Horseshoes for interval maps
	7.5 Existence of a horseshoe
8 Topological entropy
	8.1 The definition
	8.2 Independence of the metric; factor maps
	8.3 Maps on intervals and circles
	8.4 The definition with covers
	8.5 Miscellaneous results
	8.6 Positive entropy and horseshoes for interval maps
A Topology
	A.1 Elementary notions
	A.2 Compactness
	A.3 Continuous mappings
	A.4 Convergence
	A.5 Subspaces, products and quotients
	A.6 Connectedness
	A.7 Metric spaces
	A.8 Baire category
	A.9 Irreduciblemappings
	A.10 Miscellaneous results
B The Cantor set
	B.1 The construction
	B.2 Proof of Brouwer’s Theorem
	B.3 Cantor spaces
C Hints to the Exercises
Literature
Index