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Structures, i.e. symmetries over time, appear in various systems:

  • gliders in cellular automata, like Game of Life or Rule 110,
  • unmatched string's parts in rewrite systems – unchanged in multiple rewriting steps (or matched, but rendered unchanged in some part, etc.),
  • basically, each dynamic system (in the sense of chaos theory, and also wider – each system that changes its configuration over time) can have more or less symmetry.

Are there some works on the topic, i.e. on abstract treatment of symmetries in systems?

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up vote 2 down vote accepted

The topic of dynamical systems with symmetry is pretty wide. You might get a good overview by reading this non-technical paper by Mainzer.

If you want to delve into more detail, the theory of systems with spatial symmetry (as opposed to time-evolution symmetry) is explained well in the introduction to this PhD thesis by Ana Rodrigues or this book by Rececca Hoyle.

An alternate perspective is provided by the treatment of symmetry in physics. There are various profound results here. For example, Noether's theorem demonstrates that continuous symmetries correspond to conservation laws (e.g. conservation of momentum, angular momentum, energy etc).

The unification of spatial and time-evolution symmetries is also explained - in physics this normally means working with the Lorentz group or the Poincaré group, but other groups are also studied.

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