Moduli spaces for dynamical systems with portraits

Multipliers and invariants of endomorphisms of projective space in dimension greater than 1

“…These families, whose multiplier spectrums are the same for every member of the family, are called isospectral (see Definition 5.2). These results provide some partial answers to questions raised during the Bellairs Workshop on Moduli Spaces and the Arithmetic of Dynamical Systems in 2010 and subsequent notes published by Silverman [30,Question 2.43] and also raised in Doyle-Silverman [5,Question 19.5].…”

“…is the hypersurface defined by the vanishing of 36σ 5 1,2 + 4768σ 2,3 σ 3,3 − 15360σ 2,4 σ 3,3 + 1232σ 2 3,3 + 20517376σ 1,2 − 5436928σ 2,2 − 459776σ 2,3 − 1844224σ 2,4 + 532480σ 3,3 + 56702976.…”

“…Although, there were a number of closed subsets where the degree is 12. Note also that when we used only the invariants defined in Doyle-Silverman [5] ({σ (1) D 1 ,j : 1 ≤ j ≤ N D 1 }), then the computations similar to Theorem 5.19 did not finish in a reasonable amount of time even though one would still expect to find five generators and a hypersurface requirement.…”

“…This quotient exists as a geometric quotient (N = 1: [23,28], N > 1: [19,26]) and is called the moduli space of dynamical systems of degree d on P N . These moduli spaces (and their generalizations) have received much study; for example, see [3,5,21,22,23,28].…”

GIT stability of linear maps on projective space with marked points

“…We are motivated by two applications to algebraic dynamics. First, our moduli spaces are the degree d = 1 analogue of the moduli spaces M N d,n of degree d ≥ 2 dynamical systems on P N with marked points, introduced by Doyle-Silverman [7]. We show that M N 1,n exists and admits a dynamically meaningful compactification MN 1,n with subtle combinatorics at the boundary.…”