Existence of non-preperiodic algebraic points for a rational self-map of infinite order

By Ekaterina Amerik. 2010

53 citations recorded by the source corpus.

Let $X$ be a variety defined over a number field and $f$ be a dominant rational self-map of $X$ of infinite order. We show that $X$ admits many algebraic points which are not preperiodic under $f$. If $f$ were regular and polarized, this would follow immediately from the theory of canonical heights, but it does not work very well for rational self-maps. We provide an elementary proof following an argument by Bell, Ghioca and Tucker (arxiv:0808.3266).

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c4d92e6990e97093dc783482fb1343c6855c1361

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties (2012, background): “We note that Xf(K̄) always contains many points; see [2].”
Current trends and open problems in arithmetic dynamics (2018, background): “6(a) has been proven in a handful of cases [2,3,16,171], including, in particular, that Xie [244] has shown it is true for X = A(2).”
Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space (2011, background): “We remark that P(Q̄)φ is Zariski dense in P N , although the proof is not easy; see for example [3].”