In each case G is a semigroup generated by maps of maximal generic rank in (mathbb C^2).
We prove that if the Julia set of a semigroup G which is generated by endomorphisms of maximal generic rank k in (mathbb C^k) contains an isolated point, then G must contain an element that is conjugate to an upper triangular automorphism of (mathbb C^k).
As discussed in Section 1, we will be studying the properties of recurrent and wandering Fatou components of semigroup generated by entire maps of maximal generic rank on (mathbb {C}^k).
There exists different definitions of rank for tensors, like typical and generic ranks, or also symmetric rank for a symmetric tensor (see[53, 54] for more details).
Let (f_1) and (f_2) be the following maps in (mathbb C^2) of maximal generic rank: begin{aligned} F_1 z,w)= z,w^2), F_2 z,w)= z,wz).
end{aligned} Claim The generic rank of f is (r_{phi }.) By the definition of recurrence it follows that (Omega subset Omega _{phi }), where (Omega _phi) is a periodic Fatou component for (phi) with period 1. Hence by Theorem 3.3 in [5] it follows that the limit maps of the set ({phi ^n}) in (Omega _phi) have the same generic rank (say r).
(mathcal {E}_k;) The set of holomorphic endomorphisms of (mathbb C^k) which have maximal generic rank k. (mathcal {I}_k;) The set of injective holomorphic endomorphisms of (mathbb C^k).
But by the choice of h, the generic rank of (tilde{h} le r_{phi }) and (M subset tilde{h}(Omega ) cap Omega.) So the dimension of M is (r_{phi }.) Now for any point in M, the rank of the derivative matrix of (mathrm{Id}-tilde{h}) is greater than or equal to (k-r_phi).
In 1871 Paul Kummer raised most of Fries's tribes to generic rank, and so renamed the species Galorrheus volemus.
In 1890, Oryzomys was raised to generic rank, and in subsequent years numerous additional species were ascribed to it, many of which were soon moved to separate genera.
