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The dynamical degree of a dominant rational map $f:mathbb{P}^Nrightarrowmathbb{P}^N$ is the quantity $delta(f):=lim(text{deg} f^n)^{1/n}$. We study the variation of dynamical degrees in 1-parameter families of maps $f_T$. We make a conjecture and ask two questions concerning, respectively, the set of $t$ such that: (1) $delta(f_t)ledelta(f_T)-epsilon$; (2) $delta(f_t)<delta(f_T)$; (3) $delta(f_t)<delta(f_T)$ and $delta(g_t)<delta(g_T)$ for independent families of maps. We give a sufficient condition for our conjecture to hold and prove that it is true for monomial maps. We describe non-trivial families of maps for which our questions have affirmative and negative answers.
We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain infinitely many conservative maps of degree $d$; this answers a question of Silverman. Rather
We give necessary and sufficient conditions for post-critically finite polynomials to have potential good reduction at a given prime. We also answer in the negative a question posed by Silverman about conservative polynomials. Both proofs rely on dyn
Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has ADE-singularities. Let G be the reductive group given by the root system of these singularities. We construct
For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article ``Uniform Bounds on Pre-Images Under Quad
Let $Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $mathbb H$, and let $M = Gamma backslash mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $gamma$ is parabolic, there is an associated