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

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.Comment: 29 pages, 3 figure

THE PERIOD OF 2-STEP AND 3-STEP SEQUENCES IN DIRECT PRODUCT OF MONOIDS

Let M and N be two monoids consisting of idempotent elements. By the help of the presentation which defines Mx N, the period of 2-step sequences and 3-step sequences in MxN is given

Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let \UT_n(\FF_q) denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of \UT_{n+1}(\FF_q) are indexed by lattice paths from the origin to the line $x+y=n$ using the steps $(1,0), (1,1), (0,1), (1,1)$, which are labeled in a certain way by nonzero elements of \FF_q. In particular, we prove for $n\geq 1$ that the number of Heisenberg characters of \UT_{n+1}(\FF_q) is a polynomial in $q-1$ with nonnegative integer coefficients and degree $n$, whose leading coefficient is the $n$th Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of \UT_n(\FF_q) is a polynomial in $q-1$ whose coefficients are Delannoy numbers and whose values give a $q$-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of \UT_n(\FF_q) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in $q-1$ with nonnegative integer coefficients.Comment: 25 pages; v2: material significantly revised and condensed; v3: minor corrections, final versio