Piecewise isometries are dynamical systems whose behaviour
is determined by discontinuity rather than nonlinearity, resulting in striking complexity from minimal ingredients.
The dynamics -intermediate between order and chaos-
is discrete in essence, and it is effectively described
by restricting coordinates to algebraic number fields,
rather than to the familiar real or complex fields.
The resulting dynamical theory has a strong arithmetical flavour, and features prominently computer-assisted proofs.
We review recent developments.
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