Symbolic dynamics studies dynamical systems whose points are sequences of symbols from an alphabet and whose evolution is a shift. A finite alphabet with the discrete topology gives its sequence space a product topology.
A two-sided sequence over a finite alphabet is uniformly recurrent if every finite word over an alphabet occurring in it occurs with bounded gaps. More precisely, for each such word some ensures that every length- interval contains a complete occurrence. This is equivalent to being a minimal point of the full shift: a finite cover of the orbit closure by preimages of a word's cylinder set bounds its return gaps; conversely, bounded gaps pass to all points of the orbit closure and make every forward orbit dense there. The property concerns finite words, not infinite integer intervals.
The two-sided full shift on a finite alphabet consists of all functions , with the product topology and the left shift. It is a compact metric space. A compatible metric isAgreement on increasingly large finite coordinate sets is equivalent to convergence in this topology. The metric above is compatible but is not invariant under the left shift.
On a two-sided full shift, the left shift is the homeomorphismIts inverse sends to . On a one-sided sequence space, the same forward shift is generally not invertible.
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Symbolic dynamics is a branch of mathematics that studies dynamical systems through the use of symbols and sequences. It focuses on representing complex dynamical behaviors and trajectories in a simplified way using finite or countable sets of symbols. The primary idea in symbolic dynamics is to encode the states of a dynamical system as sequences of symbols. For example, one can take a continuous or discrete dynamical system and map its trajectories onto a finite alphabet (like {0, 1} for binary sequences).