Bilinear Discrete Painleve-II and its Particular Solutions

By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the $\tau$ function for d-P$\rm_{II}$. Two different forms of bilinear d-P$\rm_{II}$ are obtained and we show that they can be related by a simple gauge transformation.Comment: 9 pages in plain Te

On a q-difference Painlev\'e III equation: II. Rational solutions

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.Comment: Archive version is already official. Published by JNMP at http://www.sm.luth.se/math/JNMP

Invariant varieties of periodic points for some higher dimensional integrable maps

By studying various rational integrable maps on $\mathbf{\hat C}^d$ with $p$ invariants, we show that periodic points form an invariant variety of dimension $\ge p$ for each period, in contrast to the case of nonintegrable maps in which they are isolated. We prove the theorem: {\it `If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points of other period in the map.'}Comment: 24 page