The Moduli Space of Quadratic Rational Maps
نویسنده
چکیده
Let M2 be the space of quadratic rational maps f : P 1 → P , modulo the action by conjugation of the group of Möbius transformations. In this paper a compactification X of M2 is defined, as a modification of Milnor’s M 2 ' CP , by choosing representatives of a conjugacy class [f ] ∈ M2 such that the measure of maximal entropy of f has conformal barycenter at the origin in R, and taking the closure in the space of probability measures. It is shown that X is the smallest compactification of M2 such that all iterate maps [f ] 7→ [f ] ∈ M2n extend continuously to X → M2n , where Md is the natural compactification of Md coming from geometric invariant theory.
منابع مشابه
On the geometry of bifurcation currents for quadratic rational maps
We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1, 1)-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.
متن کاملThe Boundary of the Moduli Space of Quadratic Rational Maps
Let M2 be the space of quadratic rational maps f : P 1 → P , modulo the action by conjugation of the group of Möbius transformations. In this paper a compactification X of M2 is defined, as a modification of Milnor’s M2 ≃ CP , by choosing representatives of a conjugacy class [f ] ∈ M2 such that the measure of maximal entropy of f has conformal barycenter at the origin in R, and taking the closu...
متن کاملA Note on Quadratic Maps for Hilbert Space Operators
In this paper, we introduce the notion of sesquilinear map on Β(H) . Based on this notion, we define the quadratic map, which is the generalization of positive linear map. With the help of this concept, we prove several well-known equality and inequality...
متن کاملSpecial Curves and Postcritically Finite Polynomials
We study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier...
متن کاملA Compactification of the Moduli Space of Self-maps of Cp via Stable Maps
We present a new compactification M(d, n) of the moduli space of self-maps of CP of degree d with n markings. It is constructed via GIT from the stable maps moduli space M0,n(CP 1 ×CP, (1, d)). We show that it is the coarse moduli space of a smooth Deligne-Mumford stack and we compute its rational Picard group.
متن کامل