The dynamical structure of the rational map $ax+1/x$ on the projective line $P$ over the field $mathbb{Q}_p$ of $p$-adic numbers is described for $pgeq 3$.
In this essay we study various notions of projective space (and other schemes) over $mathbb{F}_{1^ell}$, with $mathbb{F}_1$ denoting the field with one element. Our leading motivation is the Hiden Points Principle, which shows a huge deviation betwee
n the set of rational points as closed points defined over $mathbb{F}_{1^ell}$, and the set of rational points defined as morphisms $texttt{Spec}(mathbb{F}_{1^ell}) mapsto mathcal{X}$. We also introduce, in the same vein as Kurokawa [13], schemes of $mathbb{F}_{1^ell}$-type, and consider their zeta functions.
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=F_{q}+vF_{q}+v^{2}F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construct
ion of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $F_q$ and extend these to codes over $R$.
Let $p$ be a prime and let $C_p$ denote the $p$-ary code of the projective plane over ${mathbb Z}/pmathbb{Z}$. It is well known that the minimum weight of non-zero words in $C_p$ is $p+1$, and Chouinard proved that, for $p geq 3$, the second and thir
d minimum weights are $2p$ and $2p+1$. In 2007, Fack et. al. determined, for $pgeq 5$, all words of $C_p$ of these three weights. In this paper we recover all these results and also prove that, for $p geq 5$, the fourth minimum weight of $C_p$ is $3p-3$. The problem of determining all words of weight $3p-3$ remains open.
We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form $Phi(z_1,z_2) = (phi(z_1,z_2), z_2)$. If $phi$ has degree $1$ in the first variable, the dynamics on e
ach horizontal fiber can be described in terms of Mobius transformations but the global dynamics on the $2$-torus exhibit some complexity, encoded in terms of certain $mathbb{T}^2$-symmetric polynomials. We describe the dynamical behavior of such mappings $Phi$ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.