Abstract. We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.
In this paper we study properties of endomorphisms of P k using a symmetric product construction (P 1 ) k /S k ∼ = P k . Symmetric products have been used to produce examples of endomorphisms of P k with certain characteristics, k ≥ 2. In the present note, we discuss the use of these maps to enlighten arithmetic phenomena and stability phenomena in parameter spaces. In particular, we study notions of uniform boundedness of rational preperiodic points via good reduction information, k-deep postcritically finite maps, and characterize families of Lattès maps.
In this paper we study properties of endomorphisms of
P
k
\mathbb {P}^k
using a symmetric product construction
(
P
1
)
k
/
S
k
≅
P
k
(\mathbb {P}^1)^k/\mathfrak {S}_k \cong \mathbb {P}^k
. Symmetric products have been used to produce examples of endomorphisms of
P
k
\mathbb {P}^k
with certain characteristics,
k
≥
2
k\geq 2
. In the present note, we discuss the use of these maps to enlighten stability phenomena in parameter spaces. In particular, we study
k
k
-deep post-critically finite maps and characterize families of Lattès maps.
Abstract. Let f : X X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of P 1 that is invariant under f , with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose f restricted to this line is given by z → z b , with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W s loc (S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a ≥ b cannot be relaxed without adding additional hypotheses by presenting two examples with a < b for which W s loc (S) is not real analytic in the neighborhood of any point.
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