arXiv subjects
Publications and source records attributed to Jorge Mello.
We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime $p$, extending previous work of Shparlinski (2017).
We study intersections of semigroup orbits in polynomial dynamics with multiplicative subgroups, extending results of Ostafe and Shparlinski (2010).
We study intersections of orbits in polynomial semigroup dynamics with lines on the affine plane over a number field, extending previous work of D. Ghioca, T. Tucker, M. Zieve (2008)
We give quantitative bounds for the number of quasi-integral points in orbits of semigroups of rational maps under some conditions, generalizing previous work of L. C. Hsia and J. Silverman (2011) for orbits generated by the iterations of one rational map.
We give a proof for a fact that for any Weil height $h_X$ with respect to an ample divisor on a projective variety $X$, any dynamical system $\mathcal{F}$ of rational self-maps on $X$, and any $ε>0$, there is a positive constant $C=C(X, h_X, f, ε)$ such that $\sum_{f \in \mathcal{F}_n} h^+_X(f(P)) \leq C. k^n.(δ_{\mathcal{F}} + ε)^n . h^+_X(P)$ for all points $P$ whose $\mathcal{F}$-orbit is well defined, with $δ_{\mathcal{F}}$ being a dynamical degree associated with a system of several maps, defined by the author in the previous paper mentioned above.
We prove, for the canonical height defined by Silverman [15] on monomial maps, the existence of effective lower bounds for heights of points with Zariski dense orbit, for cases with endomorphisms induced by matrices with real Jordan form.
We define arithmetical and dynamical degrees for dynamical systems with several rational maps on projective varieties, study their properties and relations, and prove the existence of a canonical height function associated with divisorial relations in the Néron-Severi Group over Global fields of characteristic zero, when the rational maps are morphisms. For such, we show that for any Weil height $h_X$ with respect to an ample divisor on a projective variety $X$, any dynamical system $\mathcal{F}$ of rational self-maps on $X$, and any $ε>0$, there is a positive constant $C=C(X, h_X, f, ε)$ such that $\sum_{f \in \mathcal{F}_n} h^+_X(f(P)) \leq C. k^n.(δ_{\mathcal{F}} + ε)^n . h^+_X(P)$ for all points $P$ whose $\mathcal{F}$-orbit is well defined.
We study families of varieties endowed with polarized canonical eigensystems of several maps, inducing canonical heights on the dominating variety as well as on the "good" fibers of the family. We show explicitely the dependence on the parameter for global and local canonical heights defined by Kawaguchi when the fibers change, extending previous works of J. Silverman and others. Finally, fixing an absolute value $v \in K$ and a variety $V/K$, we descript the Kawaguchi`s canonical local height $\hatλ_{V,E,\mathcal{Q},}(.,v)$ as an intersection number, provided that the polarized system $(V,\mathcal{Q})$ has a certain weak Néron model over Spec$(\mathcal{O}_v)$ to be defined and under some conditions depending on the special fiber. With this we extend Néron's work strengthening Silverman's results, which were for systems having only one map.