[Paper Review] The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point

Colin Christopher, Christiane Rousseau|ArXiv.org|Sep 12, 2008

Mathematical Dynamics and Fractals4 references3 citations

TL;DR

This paper establishes the complete moduli space for germs of generic one-parameter families of analytic diffeomorphisms unfolding a codimension-1 parabolic fixed point. By analyzing sectorial overlap in the $ε$-plane and comparing two representatives of the Ecalle-Voronin modulus across a full turn in the universal cover, the authors derive a necessary and sufficient compatibility condition—equivalent to 1/2-summability of the modulus in $ε$ with non-summability direction aligned to the real multipliers at the fixed points—enabling full realizability of the modulus as arising from an analytic unfolding.

ABSTRACT

In this paper we describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1. In [MRR] (and also [R]), it was shown that the Ecalle-Voronin modulus can be unfolded to give a complete modulus for such germs. The modulus is defined on a ramified sector in the canonical perturbation parameter $\eps$. As in the case of the Ecalle-Voronin modulus, the modulus is defined up to a linear scaling depending only on $\eps$. Here, we characterize the moduli space for such unfoldings by finding the compatibility conditions on the modulus which are necessary and sufficient for realization as the modulus of an unfolding. The compatibility condition is obtained by considering the region of sectorial overlap in $\eps$-space. This lies in the Glutsyuk sector where the two fixed points are hyperbolic and connected by the orbits of the diffeomorphism. In this region we have two representatives of the modulus which describe the same dynamics. We identify the necessary compatibility condition between these two representatives by comparing them both with their common Glutsyuk modulus. The compatibility condition implies the existence of a linear scaling for which the modulus is 1/2-summable in $\eps$, whose direction of non-summability coincides with the direction of real multipliers at the fixed points. Conversely, we show that the compatibility condition (which implies the summability property) is sufficient to realize the modulus as coming from an analytic unfolding, thus giving a complete description of the space of moduli.

Motivation & Objective

• To characterize the moduli space of germs of generic 1-parameter families of analytic diffeomorphisms that unfold a codimension-1 parabolic fixed point.
• To identify the necessary and sufficient conditions under which a given modulus can be realized as arising from an actual analytic unfolding.
• To resolve the open problem of realizability of the Ecalle-Voronin modulus in the context of unfoldings, particularly due to monodromy issues in the $ε$-parameter space.
• To establish that the compatibility condition derived from Glutsyuk region dynamics is equivalent to 1/2-summability of the modulus in the perturbation parameter $ε$.

Proposed method

• Analyzes the dynamics in the Glutsyuk region, where both fixed points are hyperbolic and connected by orbits, to compare two representatives of the modulus obtained by analytic continuation around $ε=0$.
• Compares the two modulus representatives via their common Glutsyuk modulus to derive a compatibility condition that must be satisfied for consistency across the Riemann surface of $ε$.
• Uses the fact that the renormalized return maps near the fixed points are linearizable in the Glutsyuk region, making the linearizing maps an analytic invariant.
• Derives the compatibility condition as a relation between the holonomy maps $γ^0$ and $γ^\infty$ associated with the two fixed points, leading to a functional equation involving $L_C$-conjugacy.
• Shows that the compatibility condition is equivalent to the modulus being 1/2-summable in $ε$, with the direction of non-summability matching the real direction of the multipliers at the fixed points.
• Proves that this summability condition is both necessary and sufficient for the modulus to be realizable as arising from an analytic unfolding, thus completing the moduli space description.

Experimental results

Research questions

• RQ1 What compatibility condition must a modulus satisfy to be realizable as the unfolding of a parabolic fixed point in a 1-parameter analytic family?
• RQ2 How does monodromy in the $ε$-parameter space affect the consistency of the Ecalle-Voronin modulus across different charts?
• RQ3 What is the precise analytic condition (in terms of summability) that ensures the modulus of an unfolding is realizable?
• RQ4 Can the moduli space of such unfoldings be completely characterized by a single analytic condition?
• RQ5 Under what conditions is the modulus representative analytic in $ε$, and how does this relate to the structure of the unfolding?

Key findings

• The compatibility condition derived from comparing modulus representatives in the Glutsyuk region is necessary and sufficient for the realizability of the modulus as arising from an analytic unfolding.
• The compatibility condition is equivalent to the modulus being 1/2-summable in the perturbation parameter $ε$, with the direction of non-summability aligned with the real direction of the multipliers at the two fixed points.
• The moduli space is fully characterized by the condition that the modulus is 1/2-summable in $ε$ with the specified non-summability direction, which ensures consistency under monodromy.
• The result justifies the terminology 'space of moduli' because the moduli depend analytically on extra parameters, and the realizability condition is fully captured by the summability property.
• In the special case where one of the $ψ^0$ or $ψ^\infty$ representatives is linear, the only realizable families are those described in Theorem 6.14, confirming a conjecture in that case.
• The solution to the functional equation arising from the compatibility condition is unique under the given constraints, and corresponds to a specific form of the holonomy map, such as $m_{(1-s)b_s, s-1}$ when $b_s$ is the first non-zero coefficient.