Variation of the canonical height in a family of rational maps

Let $d\ge 2$ be an integer, let $c(t)$ be any rational map, and let $f_t(z) := (z^d+t)/z$ be a family of rational maps indexed by t. For each algebraic number $t$, we let $h_{f_t}(c(t))$ be the canonical height of $c(t)$ with respect to the rational map $f_t$. We prove that the map $H(t):=h_{f_t}(c(t))$ (as $t$ varies among the algebraic numbers) is a Weil height

On the dynamical Bogomolov conjecture for families of split rational maps

We prove that Zhang's dynamical Bogomolov conjecture holds uniformly along $1$-parameter families of rational split maps and curves. This provides dynamical analogues of recent results of Dimitrov-Gao-Habegger and K\"uhne. In fact, we prove a stronger Bogomolov-type result valid for families of split maps in the spirit of the relative Bogomolov conjecture. We thus provide first instances of a generalization of a conjecture by Baker and DeMarco to higher dimensions. Our proof contains both arithmetic and analytic ingredients. We establish a characterization of curves that are preperiodic under the action of a non-exceptional split rational endomorphism $(f,g)$ of $(\mathbb{P}^1_{\mathbb{C}})^2$ with respect to the measures of maximal entropy of $f$ and $g$, extending a previous result of Levin-Przytycki. We further establish a height inequality for families of split maps and varieties comparing the values of a fiber-wise Call-Silverman canonical height with a height on the base and valid for most points of a non-preperiodic variety. This provides a dynamical generalization of a result by Habegger and generalizes results of Call-Silverman and Baker to higher dimensions. In particular, we establish a geometric Bogomolov theorem for split rational maps and varieties of arbitrary dimension.Comment: The proof of Theorems 4.1 and 4.3 relies on arXiv:2208.0159

Εφαρμογή της οδηγίας για τα ύδατα κολύμβησης Ευρωπαϊκή Ένωση

Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Επιστήμη και Τεχνολογία Υδατικών Πόρων

Dynamics on P1\mathbb{P}^1: preperiodic points and pairwise stability

In [DKY], it was conjectured that there is a uniform bound $B$, depending only on the degree $d$, so that any pair of holomorphic maps $f, g :\mathbb{P}^1\to\mathbb{P}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm{Rat}_d \times \mathrm{Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier-Vigny, Yuan-Zhang, and Mavraki-Schmidt. In addition, we present alternate proofs of recent results of DeMarco-Krieger-Ye and of Poineau. In fact we prove a generalization of a conjecture of Bogomolov-Fu-Tschinkel in a mixed setting of dynamical systems and elliptic curves

Height coincidences in products of the projective line

We consider hypersurfaces in $(\mathbb{P}^1)^n$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $n-1$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $n-1$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps

Fabrication and modeling of a continuous-flow microfluidic device for on-chip DNA amplification

This paper was presented at the 3rd Micro and Nano Flows Conference (MNF2011), which was held at the Makedonia Palace Hotel, Thessaloniki in Greece. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, Aristotle University of Thessaloniki, University of Thessaly, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute.The fabrication process and heat transfer computations for a continuous flow microfluidic device for DNA amplification by polymerase chain reaction (PCR) are described. The building blocks are thin polymeric materials aiming at a low cost and low power consumption device. The fabrication is performed by standard pattern transfer techniques (lithography and etching) used for microelectronics fabrication. The DNA sample flows in a meander shaped microchannel formed on a 100μm thick polyimide (PI) layer through three temperature regions defined by the integrated resistive heaters. The heat transfer computations are performed in a unit cell of the device. They show that, for the fabricated device, the variation of the temperature inside the channel zones where each step (denaturation, annealing, or extension) of PCR occur is less than 1.3K. This variation increases when the thickness of the PI layer increases. The computations also show that similar Silicon-based devices lead to lower temperature difference between the heaters and the DNA sample compared to the polymer-based fabricated device. However, the power consumption is estimated much greater for Silicon-based devices.This work was co-financed by Hellenic Funds and by the European Regional Development Fund (ERDF) under the Hellenic National Strategic Reference Framework (NSRF) 2007-2013, according to Contract no. MICRO2-45 of the Project “Microelectronic Components for Lab-on-chip molecular analysis instruments for genetic and environmental applications” within the Programme "Hellenic Technology Clusters in Microelectronics – Phase-2 Aid Measure"