Renormalization in Two-dimensional Piecewise Isometries

Piecewise-isometries are zero-entropy dynamical systems with very complex behaviour. In one-dimension they are intervalexchange transformations (IET), which are connected to diophantine arithmetic by the Boshernitzan and Carrol theorem: in any IET defined over a quadratic number field, the process of induction results, after scaling, in an eventually periodic sequence of IETs. This can be viewed as a generalization of Lagrange’s theorem on the eventual periodicity of the continued fractions of quadratic irrationals. In two dimensions we have polygon-exchange transformations (PET). Until recently, their study has been limited to specific systems defined over quadratic fields, which, invariably, have been found to exhibit eventual periodicity. I’ll describe some recent results in parametrized families of PETs, where the (fixed) rotational component identifies a quadratic field. We show that a suitable induction eventually generates a scaled version of the original map, re-parametrized by a Lüroth-type function —a piecewise affine version of Gauss’ map. The parameter values corresponding to exact scaling are found to be precisely the elements of the underlying quadratic field. The proofs required computerassistance. (Joint work with J. H. Lowenstein).