Fix integers $k,d\geq2$. Let $\operatorname{End}_d^k$ denote the parameter space of holomorphic endomorphisms of $\mathbb{P}^k$ of algebraic degree $d$. We prove that there exists a dense Zariski open subset $U_{d,k}\subset\operatorname{End}_d^k$, defined over $\mathbb{Q}$, such that, for every $f\in U_{d,k}(\mathbb{C})$, every nonconstant endomorphism $g:\mathbb{P}^k\to\mathbb{P}^k$ commuting with an iterate of $f$ is itself an iterate of $f$. An analogous dense Zariski open subset exists for regular polynomial endomorphisms of $\mathbb{C}^k$. The proof uses finite-level monodromy and its action on the rooted preimage tree of a point outside the branch locus of an iterate of $f$. As an application, we show that, for a general $f\in\operatorname{End}_d^k$, an irreducible hypersurface of $\mathbb{P}^k$ is $f$-special in the sense of Ghioca--Tucker and DeMarco--Mavraki if and only if it is $f$-preperiodic.
Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ and let $C\subseteq\mathbb{A}_{\mathbb{C}}^N$ be an irreducible curve. Suppose $C$ has an infinite intersection with the $f$-orbit of a point $x\in\mathbb{A}^N(\mathbb{C})$. Then the normalization of $C$ is isomorphic to either $\mathbb{A}^1$ or $\mathbb{...
Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the se...
Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ of algebraic degree $d$ and let $f_{\infty}$ be the induced endomorphism of the infinity hyperplane $H_{\infty}$. Suppose that for every periodic point $x_0\in H_{\infty}(\mathbb{C})$ of period $n_0$, the geometric mean of the multiplicities $e_{f_{\infty...
Let $\ell$ and $p$ be distinct prime numbers, $K$ a local field with residue characteristic $p$, $C$ a complete algebraically closed extension of $K$, and $X$ a quasi-compact quasi-separated rigid space over $K$. When $\mathop{\mathrm{char}} K = 0$, we prove that the alternating sum of traces of elements in the positiv...
Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $\delta_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)$. Then there e...
Let $K \subset \mathbb{R}^{m \times n}$ be a compact $C^1$-submanifold with boundary, $p \in (1,\infty)$ and $Q := (0,1)^n$. We prove that $K$ satisfies a rigidity estimate of the form $\|Du - (Du)_{Q}\|_{L^p} \leq C \|\mathrm{dist}_K(Du)\|_{L^p}$, $u \in W^{1,p}(Q,\mathbb{R}^m)$, if and only if $K$ satisfies sequentia...
Malte Borken· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
