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Let *A* be a finite set and consider the space
with the product topology.
The set *A* is called the alphabet, the configuration space,
and points in are called configurations.
Let act on by translation: .
A cellular automaton is a continuous map on such that
for all .
By the Curtis-Hedlund-Lyndon theorem (see e.g. [16]),
there is for every cellular automaton a finite set
such that only depends on .

A set is called invariant if for all .
A compact invariant subset of is called a subshift.
The orbit of is the set
and the orbit closure
of *x* is the closure of in .
So orbit closures are subshifts.
A subshift *X* is called minimal if for all .
A probability measure on (the Borel -algebra of)
or a subshift is called invariant if for all
measurable *G* and all .
A subshift is called uniquely ergodic if there exists only one invariant
probability measure on it; it is called strictly ergodic if it is uniquely
ergodic and minimal.

A cellular automaton maps subshifts to subshifts.
Thus the image is what is called a topological factor of
*X*.
This implies that certain properties of subshifts are invariant
under cellular automata.
Examples are: topological transitivity (i.e., there is a dense orbit),
minimality (i.e., every orbit is dense), unique ergodicity (see
Proposition 3.11 in [4]),
being of finite type, and primality (there is no nontrivial factor).

**Remarks**.

1) Prime subshifts exist [7].
If we apply a cellular automaton to a prime subshift, then
every subshift in the orbit of the cellular automaton is
topologically conjugated to the subshift
one started with unless one reaches a fixed point consisting of
a constant sequence.

2) If *X* is uniquely ergodic with invariant measure *m* and
has discrete spectrum (i.e., the eigenfunctions
are dense in , where *m* is the invariant measure),
then also has discrete spectrum
and is a subgroup of .
The reason is that the measure theoretical factor
of is isomorphic
to and that
the conditional expectation
of an eigenfunction is again an eigenfunction.

3) A cellular automaton does not increase the topological
entropy of a subshift.
This observation, however, is of limited practical
importance since for *d*=1 subshifts generically (w.r.t.\
the Hausdorff metric)
have topological entropy zero [31],
whereas in higher dimensions the topological entropy
is `as a rule' infinite (cf. [27]).
The directional topological entropy in the direction is defined
as the topological entropy of (cf. [27]).
Directional topological entropy does not increase either.

4) The fact that cellular automata leave classes of subshifts invariant
was our motivation to look at them. Call a minimal subshift *X*
palindromic, if an element (and hence all )
contains arbitrary long palindromes.
Cellular automata that commute
with the involution
map the set of palindromic subshifts into itself.
Discrete Schrödinger operators with
palindromic subshifts as potential have a generic set in their hull
for which the spectrum is purely singular continuous [19].
Therefore, applying cellular automata to palindromic subshifts
gives new classes of operators with singular continuous spectrum.

Fri Jun 26 16:29:39 CDT 1998