The objects of ergodic theory and entropy theory as outlined in the last chapters are measure preserving dynamical systems, i.e., structures with certain measurability properties. In this chapter we make additional topological assumptions on the dynamical systems, essentially that X is a compact metric space and that T acts continuously on X. There are many textbooks treating various aspects of this theory in the case d=1 (e.g., [6], [14], [60]). A good reference for more general group actions (including G = ℤd) is [42]. Interesting generalizations can be found in [45].

Pressure

For systems on finite lattices the notion of pressure was already introduced in Section 1.2. In this section we extend that notion to a broad class of dynamical systems including, in particular, the shift systems from Example 3.2.1. We start with the definition of the basic objects of our study in this chapter:

DefinitionThe pair (X, T) is called a topological dynamical system (t.d.s.), if

1. X is a compact metrizable space and

2. T is a continuous action of G = ℤd or G = ℤd on X (i.e., T9 : X → X is continuous for all g ∈ G).

In this setting we denote by B the Borel α-algebra of X and note that T acts measurably on the measurable space (X, B).

Recall that M(T) is the set of all T-invariant Borel probability measures on X and that M(T) is a convex subspace of the set M of all Borel probability measures on X.