An extremely unlikely-sounding

branch of

soft analysis, more specifically of

dynamics.

Symbolic dynamics studies a completely abstract form of a dynamical system, as will be further explained below.

But first, a rather formal definition. Take a (usually finite, always countable) set of symbols called the *alphabet*. We study two-sided infinite sequences *x*=...,*x*_{-1},*x*_{0},*x*_{1},*x*_{2},... with each *x*_{i} drawn from the alphabet. The shift operator **S** simply moves the origin:

(**S***x*)_{i} = *x*_{i+1}

Note that

**S** is an

automorphism of the dynamical system in almost all regards (it's a

homeomorphism,

invertible,

isomorphism, ...).

This gives us a trajectory for every *x*: the set of all **S**^{n}x for integers *n*.

But why is it called dynamics? Think of the symbol *x*_{i} as encoding part of the state of some ("real") system at time *i* seconds from now. Then the operator **S** is simply "wait one second". See the Kronecker system for expansion on this.