scholarly journals Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space

2012 ◽  
Vol 34 (2) ◽  
pp. 647-678 ◽  
Author(s):  
JOSEPH H. SILVERMAN
Keyword(s):  
Projective Space ◽  
Rational Map ◽  
Canonical Height ◽  
Dynamical Degree ◽  
Arithmetic Degree

AbstractLet φ:ℙN⤏ℙN be a dominant rational map. The dynamical degree of φ is the quantity δφ=lim (deg φn)1/n. When φ is defined over ${\bar {{\mathbb {Q}}}}$, we define the arithmetic degree of a point $P\in {\mathbb {P}}^N({\bar {{\mathbb {Q}}}})$ to be αφ(P)=lim sup h(φn(P))1/n and the canonical height of P to be $\hat {h}_\varphi (P)=\limsup \delta _\varphi ^{-n}n^{-\ell _\varphi }h(\varphi ^n(P))$ for an appropriately chosen ℓφ. We begin by proving some elementary relations and making some deep conjectures relating δφ, αφ(P) , ${\hat h}_\varphi (P)$, and the Zariski density of the orbit 𝒪φ(P) of P. We then prove our conjectures for monomial maps.

scholarly journals On the complex dynamics of birational surface maps defined over number fields

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Mattias Jonsson ◽  
Paul Reschke
Keyword(s):  
Number Field ◽  
Complex Dynamics ◽  
Energy Condition ◽  
Height Function ◽  
Number Fields ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Canonical Height ◽  
Dynamical Degree ◽  
Complex Projective

AbstractWe show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford–Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well behaved. We also show that there is a well-defined canonical height function.

On the degree of irrationality in Noether’s problem

2016 ◽  
Vol 12 (05) ◽  
pp. 1209-1218
Author(s):  
Jonah Leshin
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Projective Space ◽  
Function Field ◽  
Rational Map ◽  
Fixed Field ◽  
Cyclotomic Extensions ◽  
Quotient Variety ◽  
Transcendental Extension ◽  
Noether’S Problem

Noether’s problem asks whether, for a given field [Formula: see text] and finite group [Formula: see text], the fixed field [Formula: see text] is a purely transcendental extension of [Formula: see text], where [Formula: see text] acts on the [Formula: see text] by [Formula: see text]. The field [Formula: see text] is naturally the function field for a quotient variety [Formula: see text]. We study the degree of irrationality [Formula: see text] of [Formula: see text] for an abelian group [Formula: see text], which is defined to be the minimal degree of a dominant rational map from [Formula: see text] to projective space. In particular, we give bounds for [Formula: see text] in terms of the arithmetic of cyclotomic extensions [Formula: see text].

scholarly journals A rational map with infinitely many points of distinct arithmetic degrees

2019 ◽  
Vol 40 (11) ◽  
pp. 3051-3055
Author(s):  
JOHN LESIEUTRE ◽  
MATTHEW SATRIANO
Keyword(s):  
Growth Rate ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Rational Map ◽  
Arithmetic Degree

Let be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb{Q}}$. For each point $P\in X(\overline{\mathbb{Q}})$ whose forward $f$-orbit is well defined, Silverman introduced the arithmetic degree $\unicode[STIX]{x1D6FC}_{f}(P)$, which measures the growth rate of the heights of the points $f^{n}(P)$. Kawaguchi and Silverman conjectured that $\unicode[STIX]{x1D6FC}_{f}(P)$ is well defined and that, as $P$ varies, the set of values obtained by $\unicode[STIX]{x1D6FC}_{f}(P)$ is finite. Based on constructions by Bedford and Kim and by McMullen, we give a counterexample to this conjecture when $X=\mathbb{P}^{4}$.

scholarly journals Examples of dynamical degree equals arithmetic degree

2014 ◽  
Vol 63 (1) ◽  
pp. 41-63 ◽  
Author(s):  
Shu Kawaguchi ◽  
Joseph H. Silverman
Keyword(s):  
Dynamical Degree ◽  
Arithmetic Degree

scholarly journals Complex Hadamard Matrices contained in a Bose–Mesner algebra

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Takuya Ikuta ◽  
Akihiro Munemasa
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Identity Matrix ◽  
Rational Map ◽  
Absolute Value ◽  
Complex Projective ◽  
Square Matrix

AbstractAcomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH* = nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we first determine the image of a certain rational map from the d-dimensional complex projective space to C

scholarly journals CANONICAL HEIGHT FUNCTIONS FOR MONOMIAL MAPS

2013 ◽  
Vol 09 (07) ◽  
pp. 1821-1840 ◽  
Author(s):  
JAN-LI LIN ◽  
CHI-HAO WANG
Keyword(s):  
Height Function ◽  
Height Growth ◽  
Rational Maps ◽  
Rational Map ◽  
Finiteness Property ◽  
Bounded Degree ◽  
Arithmetic Complexity ◽  
Height Functions ◽  
Canonical Height

The height function measures the arithmetic complexity of a point on a variety over [Formula: see text]. The canonical height function measures the asymptotic height growth (relative to the degree growth) of a point under a dominant rational map. One property desired for the canonical height function is the Northcott finiteness property, which states that there are only finitely many points for a bounded degree and a bounded height. We show that the canonical height function for dominant rational maps does not have the Northcott finiteness property in general. We develop a new canonical height function for monomial maps. In certain cases, this new canonical height function has the desired nice properties.

scholarly journals Dynamical and arithmetic degrees for random iterations of maps on projective space

2021 ◽  
pp. 1-17
Author(s):  
WADE HINDES
Keyword(s):  
Projective Space ◽  
Random Sequence ◽  
Height Growth ◽  
Counting Problems ◽  
Dynamical Degree

Abstract We show that the dynamical degree of an (i.i.d) random sequence of dominant, rational self-maps on projective space is almost surely constant. We then apply this result to height growth and height counting problems in random orbits.

scholarly journals Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jacob L. Bourjaily ◽  
Andrew J. McLeod ◽  
Cristian Vergu ◽  
Matthias Volk ◽  
Matt von Hippel ◽  
...  
Keyword(s):  
Projective Space ◽  
Feynman Integral ◽  
Weighted Projective Space
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