Noncommutative Mathematics for Quantum Systems

136 Noncommutative Mathematics for Quantum Systems

lim sup

n→∞

1

n

logs(n,e) =lim sup

n→∞

1

n

logNk+^1 +n

=lim sup

n→∞

n+k+ 1

n

logN=logN

The fact thatkwas arbitrary implies thathtop(T) =logN.

The concept of topological entropy admits a natural extension to
that oftopological pressure, introduced by Ruelle and motivated by
questions arising in statistical mechanics.

Definition 2.2.8 Let (X,d) be a compact metric space, let
T:X→Xbe a continuous map and letf ∈C(X)be real-valued.
The topological pressure ofTatfis the numberP(T,f)defined as
follows:

P(T,f) =lim

e→ 0

lim sup

n→∞

1

n

logPn(T,f,e),

where Pn(T,f,e) = inf

{

∑x∈Fexp(−∑in=− 01 f(Ti(x)))

}

, with the

infimum taken over allFbeing(n,e)-spanning subsets forT.

The expression∑x∈Fexp(−∑ni=− 01 f(Ti(x)))is often called the
partition function. More information about topological pressure can
be found in Chapter 9 of [Wa] – note however that we are using
here rather a convention of [NS] than this of [Wa] (hence, the
minus sign before the sum in the exponent). It is easy to see that
P(T, 0) = htop(T), so that the topological pressure indeed
generalizes the topological entropy; the two concepts share several
properties.

Exercise 2.2.5 Consider the shift T on the Cantor set CN,
introduced in Theorem 2.2.4 and compute the pressurep(T, 1)and
p(T,χZ 1 )(recall the notation for the cylinder sets introduced in
Theorem 2.1.10).

2.2.4 Finite-dimensional approximations in the theory of
C∗-algebras

LetAbe a unitalC∗-algebra. We will write(φ,ψ,Mn)∈CPA(A)to
denote the facts thatφ:Mn→A,ψ:A→Mnare unital completely
positive (ucp) maps.