Stochastic Canonical Heights

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

“…We now prove that for sets of morphisms, if P ∈ P N (Q) is an almost surely wandering point, then the arithmetic degree of a P -orbit exists and is equal to the dynamical degree almost surely. To do this, we use the notion of canonical heights associated to infinite sequences γ ∈ Φ S defined in [5] and revisited in [4]. In particular, if S is height controlled, then for all points P ∈ P N (Q) and all sequences γ ∈ Φ S , the limit…”

“…Given the (nearly) uniform control over height growth rates that is possible when P is an almost surely wandering point for S (see Theorem 1.15 and Corollary 1.16 above), it is useful to have an alternative characterization of this property, which we now establish. To do this, we use the expected canonical height function E ν [ ĥ] : P N → R attached to pairs (S, ν) defined in [4,Theorem 1.2]. As we will see, this height function is a useful tool for analyzing the collective action of the maps in S on points in P N ; see, for instance, [4,Corollary 1.4].…”

“…By analogy with the well-known case of a single morphism, we will prove that the arithmetic degree of a pair (γ, P ) equals the dynamical degree δ S,ν defined in Theorem 1.3 with ν-probability one. The proof of this result uses the notion of canonical heights constructed in [5] and revisited in [4]. However, in order to define these canonical heights we must stipulate that S have further properties, which we now discuss.…”

“…In Section 3, we use canonical heights for infinite sequences of endomorphisms on P N . However, it is worth mentioning that the constructions in [4,5] that we use in Section 3 work for polarizable sets of endomorphisms on more general projective varieties.…”

“…and a characterization of its kernel [4, Corllary 1.4]); see [4] or Section 4 for more details. To state this recharacterization, we define the (total) S-orbit of a point Q ∈ P N (Q) S to be the union of all possible orbits defined in (1), namely (5) Orb…”