Regular polynomial automorphisms in the space of planar quadratic rational maps
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Journal of Fixed Point Theory and Applications
Regular polynomial automorphisms in the space of planar quadratic rational maps Hyejin Kwon, Chong Gyu Lee
and Sang-Min Lee
Abstract. In this paper, we describe the semistable quotient of the set of regular polynomial automorphisms H22 in the semistable locus of the moduli space of quadratic rational maps, using the portrait moduli space of rational maps with a fixed point. We also provide the parametrization of H22 using two invariants, det f and tr f . Mathematics Subject Classification. Primary 11G50, 37P30; Secondary 14G50, 32H50, 37P05. Keywords. Polynomial automorphism, H´enon map, fixed point, moduli space, portrait.
1. Introduction A regular polynomial automorphism is an algebraically stable polynomial map equipped with the inverse which is also a polynomial map (we refer [4–6] for the definition and basic properties of the regular polynomial automorphism to the reader.) It has been spotlighted in the study of arithmetic and algebraic dynamics, especially when we test theories in dynamics for general rational maps. They share many properties with endomorphisms, which make us to study dynamical phenomena conveniently. For example, they have preperiodic points of bounded height [1,11], a canonical height [3–6,13], and equidistributed periodic points with respect to the dynamical invariant measure [7]. The set of regular polynomial automorphisms sits inside the parameter space Ratnd of degree d rational maps on Pn . Since the natural conjugation action of PGLn+1 respects geometric properties of dynamical systems defined by self maps on Pn , we take the quotient space of Ratnd by PGLn+1 and take a representative of each equivalent classes for computational convenience. Also, we are interested in geometric properties of the quotient space itself. It is This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education. (NRF2016R1D1A1B01009208). 0123456789().: V,-vol
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known that the conjugacy classes of endomorphisms forms a rational variety in Mnd [9]. We also have the description of the moduli space of quadratic rational maps on P1 [12,14] and the moduli space of quadratic rational maps on P1 with level structure [10]. In this paper, we focus on regular polynomial automorphisms of degree 2 on A2 defined over an algebraically closed field K of characteristic 0. We find that they are conjugate to H´enon maps: fP = (y, Dx + P (y)),
P (y) ∈ K[y]
(Theorems 3.3, 3.5), and hence, they are semistable [8, Theorem 1.2(c)]. Therefore, we can consider the semistable quotient of the space of regular polynomial automorphisms: H22 (K) := f : A2 → A2 | f : regular polynomial automorphism defined over K, deg f = 2} // PGL3 (K). We denote an element of H22 (K), which is the conjugacy class of f by PGL3 (K), with [f ] for notational convenience. In particular, we parametrize it using invariants of regular polynomial automorphisms. We define the determinant and the t
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