Narrow Results

We study the dissipative Hofstadter model on a triangular lattice, making use of the $O(2,2;R)$ T-dual transformation of string theory. The $O(2,2;R)$ dual transformation transcribes the model in a commutative basis into the model in a noncommutative basis. In the zero-temperature limit, the model exhibits an exact duality, which identifies equivalent points on the two-dimensional parameter space of the model. The exact duality also defines magic circles on the parameter space, where the model can be mapped onto the boundary sine-Gordon on a triangular lattice. The model describes the junction of three quantum wires in a uniform magnetic field background. An explicit expression of the equivalence relation, which identifies the points on the two-dimensional parameter space of the model by the exact duality, is obtained. It may help us to understand the structure of the phase diagram of the model.

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring $D[x,\sigma ,\delta ]$. We extend results of the first author and Vishkautsan on polynomial dynamics in $D[x]$. In particular, we show that if $a\in D$ and $f\in D[x,\sigma ,\delta ]$ satisfy $f(a)=a$, then $f^{\circ n}(a)=a$ for every formal power of $f$. More generally, we give a sufficient condition for a point $a$ to be $r$-periodic with respect to a polynomial $f$. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.