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Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a…

Given an algebraic function field $F|K$ and a place $\wp$ on $K$, we prove that the places that are composite with extensions of $\wp$ to finite extensions of $K$ lie dense in the space of all places of $F$, in a strong sense. We apply the…

The present note studies \emph{surjective rational endomorphisms} $f: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ with \emph{cubic} terms and the indeterminacy locus $I_f \ne \emptyset$. We develop an experimental approach, based on some…

In this paper, we study conditions for the existence of an embedding $\widetilde{f} \colon P \to Q \times \mathbb{R}$ such that $f = \mathrm{pr}_Q \circ \widetilde{f}$, where $f \colon P \to Q$ is a piecewise linear map between polyhedra.…

Let $S \subset \P^n$ be a smooth quartic hypersurface defined over a number field $K$. If $n \ge 4$, then for some finite extension $K'$ of $K$ the set $S(K')$ of $K'$-rational points of $S$ is Zariski dense.

We show that in the family of degree $d\geq 2$ rational maps of the Riemann sphere, the closure of strictly postcritically finite maps contains a (relatively) Baire generic subset of maps displaying maximal non-statistical behavior: for a…

This will is an expository description of quadratic rational maps. Sections 2 through 6 are concerned with the geometry and topology of such maps. Sections 7--10 survey of some topics from the dynamics of quadratic rational maps. There are…

We develop an abstract model for the dynamics of an exponential map $z\mapsto \exp(z)+\kappa$ on its set of escaping points and, as an analog of Boettcher's theorem for polynomials, show that every exponential map is conjugate, on a…

We prove in this article the surjectivity of three maps. We prove in Theorem $1.6$ the surjectivity of the Chinese remainder reduction map associated to the projective space of an ideal with a given factorization into ideals whose radicals…

A Peano compactum is a compact metric space having locally connected components such that at most finitely many of them are of diameter greater than any fixed number C>0. Given a compactum K in the extended complex plane, it is known that…

We prove that arboreal Galois extensions of number fields are never abelian for post-critically finite rational maps and non-preperiodic base points. For polynomials, this establishes a new class of known cases of a conjecture of…

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…

We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. A rational map $\phi\in K(z)$ of degree at least $2$ is subhyperbolic if each critical point in the $\mathbb{C}_p$-Julia set of $\phi$ is eventually periodic. We…

We consider the Zariski space of all places of an algebraic function field $F|K$ of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime…

For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational…

For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher…

A continous map $f: \mathbb{C}^n \rightarrow \mathbb{C}^N$ is $k$-regular if the image of any $k$ points spans a $k$-dimensional subspace. It is an important problem in topology and interpolation theory, going back to Borsuk and Chebyshev,…

In this article, we prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps $f_t: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$, parameterized by $t$ in a…

Let $\mathcal{C}(S^{m})$ denote the set of continuous maps from the unit sphere $S^{m}$ in $\mathbb{R}^{m+1}$ into itself endowed with the supremum norm. We prove that the set $\{f^n: f\in \mathcal{C}(S^{m})~\text{and}~n\ge 2\}$ of iterated…