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

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.

A central limit theorem for the degree of a random product of Cremona transformations

“…In other words, the sequence log deg(f n ) follows a law of large numbers. This was shown in the case of rational maps on P k in [Hin19], where the quantity ℓ µ is referred to as the random dynamical degree. For birational maps of P 2 , it was shown in [MT18b] that the limit ℓ µ is positive if the semigroup generated by the support of µ is non-elementary.…”

Orbit counting in polarized dynamical systems

“…For instance, if X is a set or rational or integral points, then this growth rate is known to encode many interesting arithmetic and geometric invariants of V (like its dimension, genus, or rank of an associated Mordell-Weil group); for examples, see [2,8,10,11,20]. Likewise, when X is a dynamical orbit generated by a collection of self maps of V , then the growth rate on the number of points in X of bounded height frequently detects dynamical degrees [14,17] as well as other invariants [1,21]. In this paper, we take up this dynamical orbit counting problem, generalizing the main results from [13] in two ways: first we allow infinitely generated semigroups, and second we allow V to be any projective variety with a polarizable set of maps (not necessarily P N ) .…”

Orbit counting in polarized dynamical systems

“…For instance, if X is a set or rational or integral points, then this growth rate is known to encode many interesting arithmetic and geometric invariants of V (like its dimension, genus, or rank of an associated Mordell-Weil group); for examples, see [2,8,10,11,20]. Likewise, when X is a dynamical orbit generated by a collection of self maps of V , then the growth rate on the number of points in X of bounded height frequently detects dynamical degrees [14,17] as well as other invariants [1,21]. In this paper, we take up this dynamical orbit counting problem, generalizing the main results from [13] in two ways: first we allow infinitely generated semigroups, and second we allow V to be any projective variety with a polarizable set of maps (not necessarily P N ) .…”

“…Remark 1. The first condition, which we call height controlled but has also been called bounded, has been used in several places to study both semigroup and random sequential orbits; see, for instance, [12,14,16,19]. However the second condition is a new and technical one, allowing us to control the dominant poles of some associated meromorphic generating functions; see Section 2 for details.…”