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,x0,x1,x2,... with each xi drawn from the alphabet. The shift operator S simply moves the origin:

(Sx)i = xi+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 Snx for integers n.

But why is it called dynamics? Think of the symbol xi 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.

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