2022/02/13 by Jason P. Bell, Dragos Ghioca, Bell, Jason +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2202.06364
We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety X defined over an algebraically closed field K of characteristic 0, endowed with a birational self-map ϕ of dynamical degree 1, we expect that either there exists a non-constant rational function f:X\dashrightarrow ℙ1 such that f∘ ϕ=f, or there exists a proper subvariety Y⊂ X with the property that for any invariant proper subvariety Z⊂ X, we have that Z⊆ Y. We prove our conjecture for automorphisms ϕ of dynamical degree 1 of semiabelian varieties X. Also, we prove a related result for regular dominant self-maps ϕ of semiabelian varieties X: assuming ϕ does not preserve a non-constant rational function, we have that the dynamical degree of ϕ is larger than 1 if and only if the union of all ϕ-invariant proper subvarieties of X is Zariski dense. We give applications of our results to representation theoretic questions about twisted homogeneous coordinate rings associated to abelian varieties.