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We study various aspects of the dynamics induced by integer matrices on theinvariant rational lattices of the torus in dimension 2 and greater. Firstly,we investigate the orbit structure when the toral endomorphism is notinvertible on the lattice, characterising the pretails of eventually periodicorbits. Next we study the nature of the symmetries and reversing symmetries oftoral automorphisms on a given lattice, which has particular relevance to(quantum) cat maps.

Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices