scholarly journals Overconvergent subanalytic subsets in the framework of Berkovich spaces

2016 ◽  
Vol 18 (10) ◽  
pp. 2405-2457 ◽  
Author(s):  
Florent Martin
Keyword(s):  
Berkovich Spaces

scholarly journals Berkovich spaces embed in Euclidean spaces

2014 ◽  
Vol 60 (3) ◽  
pp. 273-292 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser ◽  
Bjorn Poonen
Keyword(s):  
Euclidean Spaces ◽  
Berkovich Spaces

scholarly journals On the distribution of orbits in affine varieties

2014 ◽  
Vol 35 (7) ◽  
pp. 2231-2241
Author(s):  
CLAYTON PETSCHE
Keyword(s):  
Ergodic Theory ◽  
Closed Subset ◽  
Arithmetic Progressions ◽  
Affine Variety ◽  
Arbitrary Characteristic ◽  
Berkovich Spaces ◽  
Density Zero ◽  
Banach Density

Given an affine variety $X$, a morphism ${\it\phi}:X\rightarrow X$, a point ${\it\alpha}\in X$, and a Zariski-closed subset $V$ of $X$, we show that the forward ${\it\phi}$-orbit of ${\it\alpha}$ meets $V$ in at most finitely many infinite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the dynamical Mordell–Lang conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces.

Applications to the topology of Berkovich spaces

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Closed Field ◽  
Projective Varieties ◽  
Homotopy Types ◽  
Berkovich Spaces ◽  
Galois Orbits ◽  
Finite Simplicial Complex ◽  
Upper Level ◽  
Definable Functions ◽  
Strong Deformation Retraction ◽  
Deformation Retraction

This chapter presents various applications to the topology of classical Berkovich spaces. It deduces from the main theorem several new results on the topology of V(superscript an) which were not known previously in such a level of generality. In particular, it shows that V(superscript an) admits a strong deformation retraction to a subspace homeomorphic to a finite simplicial complex and that V(superscript an) is locally contractible. The chapter also proves the existence of strong retractions to skeleta for analytifications of definable subsets of quasi-projective varieties and goes on to prove finiteness of homotopy types in families in a strong sense and a result on homotopy equivalence of upper level sets of definable functions. Finally, it describes an injection in the opposite direction (over an algebraically closed field) which in general provides an identification between points of Berkovich analytifications and Galois orbits of stably dominated points.

Sign in / Sign up

Export Citation Format

Share Document