Noncommutative Mathematics for Quantum Systems

Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 135

(and Sinai) in 1950s for measurable dynamical systems, but later
defined also for topological dynamical systems by Adler,
Konheim, and McAndrew (see [NS] and references therein). Here
we will recall an equivalent definition in the context of metric
spaces, proposed by Bowen.
Let(X,d)be a compact metric space and letT :X → Xbe a
continuous map. A finite setF⊂⊂Xis called(n,e)-spanningforT
(n∈N,e>0) if

∀x∈X∃f∈F∀k=0,...,n− 1 d(Tkx,Tkf)<e.

(n,e)-spanning sets always exist (Exercise! – it follows from
compactness ofXand continuity ofT). Define

s(n,e) =min{card(F):F−(n,e)−spanning subset forT}.

Definition 2.2.6 Topological entropy of T is the real number
(possibly 0, possibly+∞)

htop(T) =sup

e> 0

lim sup

n→∞

1

n

logs(n,e)

(in fact, one can replace in the definition the lim sup by lim inf, see
Chapter 7 in [Wa]).

The idea behind Bowen’s definition is related to approximating
a given dynamical system (in the firstnsteps, up toe)by a finite
system. Below we will apply a similar scheme to define entropy
for quantum dynamical systems. Before we do that let us first,
however, compute the entropy of the one-sided shift on the Cantor
set.

Proposition 2.2.7 LetTbe the shift onCNintroduced in Theorem
2.2.4. Thenhtop(T) =logN.

Proof Fix for the momentk∈Nand pute=^1 k. It is easy to check
thatFk⊂⊂CNis ane-net if and only if for any wordwof length
k+1 (built of letters from the alphabet{1,... ,N}) one can find a
word inFksuch that its initial(k+ 1 )-long segment coincides with
w. Thus, the minimal cardinality of ane-net is equal toNk+^1. A
similar argument shows that for anyn∈Nthe minimal cardinality
of an(n,e)-spanning set forTis equal toNk+^1 +n. Thus we have