We let $𝕃$ be the completion of the field of formal Puiseux series and study polynomials with coefficients in $𝕃$ as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in $𝕃\left[\zeta \right]$. We show that cubic polynomial dynamics over $𝕃$ and $ℂ$ are intimately related. More precisely, we establish that some elements of $𝕃$ naturally correspond to the Fourier series of analytic almost periodic functions (in the sense of Bohr) which parametrize (near infinity) the quasiconformal...

Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\to Y$ be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{♯}$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus foranycurrent, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can...

We study the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties. Our method is based on the intersection theory on toric varieties. We use the method to determine the dynamical degrees of monomial maps and compute the degrees of the Cremona involution.