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Contracting rigid germs in higher dimensions. (Germes rigides contractants en toute dimension.) (English. French summary) Zbl 1316.37025

The author presents normal forms for non-invertible contracting rigid germs in several complex variables.
Poincaré-Dulac normal forms are an example of such classifications in terms of analytic and formal conjugacy, valid for locally invertible germs. Those depend on the linear part of the germ at the origin and may feature resonant monomials. See also [S. Sternberg, Am. J. Math. 79, 809–824 (1957; Zbl 0080.29902)], [J.-P. Rosay and W. Rudin, Trans. Am. Math. Soc. 310, No. 1, 47–86 (1988; Zbl 0708.58003)] and [F. Berteloot, Ann. Fac. Sci. Toulouse, Math. (6) 15, No. 3, 427–483 (2006; Zbl 1123.37019)] for the normal forms of locally invertible contracting germs.
A contracting germ \(f\) is a germ for which all the eigenvalues of the linear part have a modulus less than \(1\). The generalized critical set is defined as the union of all the critical sets of the iterates of \(f\). In the non-invertible case, the topological structure of the generalized critical set appears as a natural formal invariant.
A contracting rigid germ is then a contracting germ for which the generalized critical set has simple normal crossing singularities at the origin, i.e., the irreducible components of the generalized critical set can be represented by the equations \(w^k=0\), \(k=1,\dots,q\) for a choice of coordinates \((w_1,\dots,w_d)\).
C. Favre’s work [J. Math. Pures Appl. (9) 79, No. 5, 475–514 (2000; Zbl 0983.32023)] gives a classification for contracting rigid germs in dimension \(2\). This consists of polynomial normal forms.
The author’s work is an attempt to generalize this classification. His main result, Theorem A, is the following:
Any contracting rigid germ in dimension \(d\geq 2\) with injective internal action has an iterate which is holomorphically conjugated to the following normal form, for which \(\mathbb{C}^d\) is splitted in three subspaces: \[ (x,y,z)\in\mathbb{C}^d\mapsto(\sigma_{PD}(x), \beta x^E y^D (\mathbf{1} + g(x)), h(x, y, z)) \] where
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\(\sigma_{PD}\) is a Poincaré-Dulac normal form,
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\(\beta x^E y^D\) is a vector valued monomial,
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\(g\) is a vector-valued polynomial,
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\(h\) is a holomorphic function with vanishing derivative at \(0\).
The internal action of a contracting rigid germ is the linear action induced on the set of irreducible components of the generalized critical set by pull-back. In other words, this is the action of the germ on the fundamental group of the complement of the generalized critical set in a small polydisk.
Note that, as remarked by the author, the theorem is also valid for general complete metrized field of zero characteristic provided the eigenvalues belong to the field. Also, examples show that this is not valid in positive characteristic.
Due to the stability of the \(x\)- and \((x,y)\)-spaces under \(f\), a corollary of this theorem (Corollary B) is the existence of a finite decreasing sequence of smooths foliations preserved by the action of \(f\).
In the case of contracting rigid germs, not only classical resonances play a role in the normal form, but also another type of resonances, called secondary resonances, which are related to the eigenvalues of the internal action.
The proof of the theorem is by successive reductions of the form. Each reduction has two steps: its proof in terms of formal series and the study of the convergence properties of the formal series.
After proving Theorem A, the authors gives the resulting (incomplete) classifications for contracting rigid germs with injective internal action in dimension \(3\). Some cases of the classification need further investigation. Indeed, the author gives examples of germs in dimension \(3\) for which the classification does not give full understanding. These examples hint that the geometry of sets related to the generalized critical set might play a role in the understanding of those exceptional cases.

MSC:

37F25 Renormalization of holomorphic dynamical systems
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References:

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