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Martin Klimes

Publications and source records attributed to Martin Klimes.

9 recordsLinked to original sources
arXiv2021

Reading analytic invariants of parabolic diffeomorphisms from their orbits

In this paper we study germs of diffeomorphisms in the complex plane. We address the following problem: How to read a diffeomorphism $f$ knowing one of its orbits $\mathbb{A}$? We solve this problem for parabolic germs. This is done by associating to the orbit ${\mathbb{A}}$ a function that we call the dynamic theta function $\Theta_{\mathbb{A}}$. We prove that the function $\Theta_{\mathbb{A}}$ is $2\pi i\mathbb{Z}$-resurgent. We show that one can obtain the sectorial Fatou coordinate as a Laplace-type integral transform of the function $\Theta_{\mathbb{A}}$. This enables one to read the analytic invariants of a diffeomorphism from the theta function of one of its orbits. We also define a closely related fractal theta function $\tilde{\Theta}_{\mathbb{A}}$, which is inspired by and generalizes the geometric zeta function of a fractal string, and show that it also encodes the analytic invariants of the diffeomorphism.

math.DS↗
arXiv2021

A remark on first integrals of vector fields

We provide examples of vector fields on $(\mathbb{C}^3, 0)$ admitting a formal first integral but no holomorphic first integral. These examples are related to a question raised by D. Cerveau and motivated by the celebrated theorems of Malgrange and Mattei-Moussu.

math.CA↗
arXiv2021

The wild monodromy of the Fifth Painlevé equation and its action on wild character variety: an approach of confluence

The article studies the Fifth Painlevé equation and of the nonlinear Stokes phenomenon at its irregular singularity at infinity from the point of view of confluence from the Sixth Painlevé equation. This approach is developped separately on both sides of the Riemann-Hilbert correspondance. On the side of the nonlinear Painlevé-Okamoto foliation the relation between the nonlinear monodromy group of Painlevé VI and the "nonlinear wild monodromy pseudogroup" of Painlevé V (that is the pseudogroup generated by nonlinear monodromy, nonlinear Stokes operators and nonlinear exponential torus) is explained in detail. On the side of the corresponding linear isomonodromic problem, the "wild" character variety (the space of the linear monodromy and Stokes data) associated to Painlevé V is constructed through a birational transformation from the character variety (the space of the linear monodromy data) associated to Painlevé VI. This allows to transport the known description of the action of the nonlinear monodromy of Painlevé VI on its character variety to that of Painlevé V, and to provide explicit formulas for the action of the "nonlinear wild monodromy" of Painlevé V on its character variety.

math.CA↗
arXiv2020

On the universal unfolding of vector fields in one variable: A proof of Kostov's theorem

In this note we present variants of Kostov's theorem on a versal deformation of a parabolic point of a complex analytic $1$-dimensional vector field. First we provide a self-contained proof of Kostov's theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the $C^\infty$ case, where we show that only versality is possible.

math.DS↗
arXiv2017

Generic 2-parameter perturbations of parabolic singular points of vector fields in C

We describe the equivalence classes of germs of generic $2$-parameter families of complex vector fields $\dot z = ω_ε(z)$ on $\mathbb{C}$ unfolding a singular parabolic point of multiplicity $k+1$: $ω_0= z^{k+1} +o(z^{k+1})$. The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fields $\dot z = z^{k+1} + ε_1 z + ε_0$ over $\mathbb{CP}^1$. This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) unique normal forms for the equivalence classes of germs.

math.DS↗
arXiv2017

Stokes phenomenon and confluence in non-autonomous Hamiltonian systems

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painleve equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed. Key words: Non-autonomous Hamiltonian systems; irregular singularity; non-linear Stokes phenomenon; wild monodromy; confluence; local analytic classification; Painleve equations.

math.CA↗
arXiv2015

Confluence of singularities in hypergeometric systems

A system in a Birkhoff normal form with an irregular singularity of Poincare rank 1 at the origin and a regular singularity at infinity is through the Borel-Laplace transform dual to a system in an Okubo form. Schafke has showed that the Birkhoff system can also be obtained from the Okubo system by a simple limiting procedure. The Okubo system comes naturally with two kinds of mixed solution bases, both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin. One can then define Stokes matrices of the Okubo system as connection matrices between different branches of the mixed solution bases and use them to relate the monodromy matrices of the Okubo system to the usual Stokes matrices of the limit system at the irregular singularity. This is illustrated on the example of confluence in the generalized hypergeometric equation.

math.CA↗
arXiv2015

Confluence of singularities of non-linear differential equations via Borel--Laplace transformations

Borel summable divergent series usually appear when studying solutions of analytic ODE near a multiple singular point. Their sum, uniquely defined in certain sectors of the complex plane, is obtained via the Borel--Laplace transformation. This article shows how to generalize the Borel--Laplace transformation in order to investigate bounded solutions of parameter dependent non-linear differential systems with two simple (regular) singular points unfolding a double (irregular) singularity. We construct parametric solutions on domains attached to both singularities, that converge locally uniformly to the sectoral Borel sums. Our approach provides a unified treatment for all values of the complex parameter.

math.CA↗