Propagation of Zariski Dense Orbits

2023/07/22 by Pasten, Hector, Silverman, Joseph H.
#37P30 #37P55 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Primary: 37P15 #Secondary: 37P05

paper · doi:10.48550/arxiv.2307.12097

Abstract

Let X/K be a smooth projective variety defined over a number field, and let f:X→X be a morphism defined over K. We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point P₀∈X(K) whose f-orbit O_f(P₀):=\fⁿ(P):n∈ℕ\ is Zariski dense, then there are many such points. For example, a weak conclusion would be that X(K) is not the union of finitely many (grand) f-orbits, while a strong conclusion would be that any set of representatives for the Zariski dense grand f-orbits is Zariski dense. We prove statements of this sort for various classes of varieties and maps, including projective spaces, abelian varieties, and surfaces.

Related

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