Survival of Arnold's invariant manifold at a transversal one-fold resonance crossing in the Siegel domain

For an analytic family of complex vector fields $\dot{\mathbf{z}} = \mathbf{a}(\mathbf{z},\varepsilon)$ whose linear-part eigenvalues transversally cross a one-fold resonance at $\varepsilon = 0$, Arnold conjectured in 1969 that a codimension-1 invariant manifold $M_\varepsilon$ exists for almost all families, given in first approximation by $F = c_0\varepsilon w_1 + c_1 w^m$, and degenerating at $\varepsilon = 0$ into the separatrix (resonant cone) manifold $M_0$. The conjecture was proven only in the Poincaré domain (Brushlinskaya; Pyartli for resonances $m_1\lambda_1+m_2\lambda_2=0$ under a strong restriction on the remaining eigenvalues), while in the Siegel domain the normal-form series diverge for most eigenvalues and the fate of $M_0$ was unknown. The later literature resolves the naive analytic statement with a precise dichotomy: Bruno (1974) showed the formal integral manifold is not always analytic, while Stolovitch (1994) proved its analyticity in any dimension under Bruno's arithmetic condition $(\omega)$ on the eigenvalues and showed that in dimension $n > 2$ some arithmetic condition on the eigenvalues is necessary for this analytic conclusion, explicitly leaving open whether $(\omega)$ itself is necessary; Garay and van Straten (2020) recovered a modified version in which the family of invariant manifolds is Whitney-$C^\infty$ rather than analytic over a positive-measure parameter set. What remains open is the arithmetic-free picture in the Siegel domain: the exact boundary between Bruno's divergent regime and Stolovitch's convergent regime, and what object, if any, replaces $M_\varepsilon$ beyond it.

Background

A holomorphic (complex analytic) vector field germ $\dot{\mathbf{z}} = A\mathbf{z} + \cdots$ with an isolated singular point at the origin is classified by its spectrum: the eigenvalues $\lambda = (\lambda_1,\dots,\lambda_n)$ of the linear part. The Poincaré domain $\Pi$ is the set of spectra whose convex hull in $\mathbb{C}$ does not contain $0$; its complement, the Siegel domain $S$, is where small-divisor problems concentrate. A resonance is an integer relation $(\mu,\lambda) := \sum_j \mu_j\lambda_j = 0$ with $\mu \in \mathbb{Z}^n$; resonant monomials $\mathbf{z}^\mu$ cannot be removed by the Poincaré–Dulac normal-form reduction. Inside a resonant hyperplane $\Gamma_\mu$, a spectrum has a one-fold (simple) resonance if it satisfies exactly one independent integer relation; this is a complex codimension-1 condition, so a generic one-parameter analytic family $\mathbf{a}(\mathbf{z},\varepsilon)$ crosses it transversally. In the Poincaré domain the resonant planes are discrete and nonresonant terms can be eliminated analytically (Poincaré); in the Siegel domain they are dense, and by Siegel's and Bruno's results the normalizing series converge only for spectra satisfying arithmetic (Diophantine) conditions, diverging generically. When a family crosses a one-fold resonance, the resonant monomial $w = \mathbf{z}^{\mu^+}$ together with $w_1 = \mathbf{z}^{\mu^-}$ ($\mu = \mu^+ - \mu^-$) generates a one-dimensional cocycle, governed by the reduced dynamics $\dot\varphi = \varphi\,(c_0(\varepsilon) + c_1(\varepsilon)\varphi)$, where $c_0(\varepsilon) = (\mu,\lambda(\varepsilon))$ measures the distance to resonance; the nonzero fixed point $\varphi = -c_0(\varepsilon)/c_1(\varepsilon)$ of this reduced dynamics corresponds to an invariant hypersurface, exactly as a limit cycle corresponds to a fixed point of a Poincaré map. Arnold's 1969 hypothesis was that for almost all such families this cocycle has an actual (analytic) fixed point: a codimension-1 invariant manifold $M_\varepsilon$ given in first approximation by $F = c_0\varepsilon w_1 + c_1 w^m$, degenerating at $\varepsilon = 0$ into the separatrix manifold $M_0$ — the resonant cone — of the resonant member of the family; the real form of the simplest case $\lambda_1 = -\lambda_2$ is the classical Hopf bifurcation of a limit cycle with radius $|\varepsilon|^{1/2}$. In the Poincaré domain this was proven by Brushlinskaya, and the intersection of the phase foliation with a small sphere is transversal, so the local topology is conical over a foliation of $S^{2n-1}$. In the Siegel domain the higher-order terms that cannot be eliminated are only known to be pushable to arbitrarily high degree, and the conjectured survival of $M_0$ under them — the analyticity of the resonant cone and of its deformation $M_\varepsilon$ — was left open; the topological restructuring was treated by Arnold only in the single two-dimensional example $\lambda_1 = 2\lambda_2$ (bifurcation of a double periodic trajectory on $S^3$).

Problem Statement

Let $\dot{\mathbf{z}} = \mathbf{a}(\mathbf{z},\varepsilon)$ be an analytic family of holomorphic vector fields with a nondegenerate singular point at the origin whose distinct, nonzero eigenvalues $\lambda(\varepsilon)$ cross a one-fold resonance $(\mu,\lambda) := \sum_j \mu_j \lambda_j = 0$ for some integer multi-index $\mu \neq 0$ transversally at $\varepsilon = 0$, the spectrum being otherwise generic. Arnold's 1969 hypothesis asserts that for almost all such families there exists a codimension-1 invariant manifold $M_\varepsilon$ which, after a change of variables bianalytic in $\mathbf{z}$ and analytic in $\varepsilon$, is given by $F(\varepsilon,\mathbf{w}) = 0$ with Taylor expansion beginning $F = c_0\varepsilon w_1 + c_1 w^m$ (where $\mathbf{w}$ are the coordinates after the change of variables, $w_1 = \mathbf{z}^{\mu^-}$ the second monomial and $w = \mathbf{z}^{\mu^+}$ the resonant monomial of the Background, so that $w^m$ is the $m$-th power of the latter, and $c_0, c_1$ are nonzero coefficients, the leading coefficients of the functions $c_0(\varepsilon) = (\mu,\lambda(\varepsilon))$ and $c_1(\varepsilon)$ of the Background (so $c_0(\varepsilon) = c_0\varepsilon + O(\varepsilon^2)$)), and which degenerates at $\varepsilon = 0$ into the separatrix manifold (resonant cone) $M_0$ of the resonant member of the family; the associated one-dimensional cocycle, governed by the reduced dynamics $\dot\varphi = \varphi(c_0(\varepsilon) + c_1(\varepsilon)\varphi)$, exists in general only as a formal series.

The question is: in the Siegel domain, where the normal-form series diverge for most spectra, determine the exact boundary between the regime in which the formal integral manifold (and the resonant cone $M_0$) is analytic and Bruno's divergent regime. The known sides of this boundary are: sufficient — Bruno's arithmetic condition $(\omega)$ on the eigenvalues (Stolovitch 1994), namely $\omega(p) := \min_{1 \leq i \leq n} \min\{\, |(Q,\lambda) - \lambda_i| \,:\, 2 \leq |Q| \leq p,\ (Q,\lambda) \neq \lambda_i \,\}$ over a strictly increasing sequence $p_k$ with $\sum_k 1/p_k < +\infty$, requiring $\sum_k -\frac{1}{p_k}\ln \omega(p_k) < +\infty$; Diophantine lower bounds $|(\mu,\lambda)| \geq \gamma|\mu|^{-\tau}$ for all $\mu \neq 0$ and some $\gamma,\tau > 0$ imply $(\omega)$, which is strictly weaker. Necessary — in dimension $n > 2$, some arithmetic condition on the eigenvalues is necessary for the analytic partial normal form exhibiting the invariant varieties (Stolovitch 1994, §3.2, proved through a dimension-3 counterexample strongly violating $(\omega)$, within the class of fixed vector fields whose resonances are generated by a single integer vector $r \in \mathbb{N}^n$); Stolovitch explicitly does not prove that $(\omega)$ itself is necessary. Negative — Bruno's non-analytic examples. In particular, the question has two parts, both required and independent: first, whether the necessary and sufficient arithmetic conditions coincide — that is, whether $(\omega)$ is necessary and, if not, which weaker arithmetic condition replaces it as the boundary; second, what invariant object, if any, replaces the analytic family $M_\varepsilon$ for spectra outside the convergent regime.

References

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