arXiv subjects
Publications and source records attributed to Nataliya Goncharuk.
We construct hyperbolic horseshoes for piecewise-analytic homeomorphisms of the circle with {\it multiple} break-type singularities, under the assumptions of bounded type rotation numbers and bounded geometry -- provided the sizes of the breaks are uniformly small. As a consequence, we obtain a $C^{1+α}$-rigidity result for such maps and prove that rigidity classes are analytic submanifolds.
We construct hyperbolic renormalization horseshoes for analytic generalized interval exchange transformations (GIETs) of bounded type and with bounded geometry, provided their boundaries are small (but not necessarily zero). As a consequence, we describe the manifold structure of smooth conjugacy classes of GIETs.
We present a novel approach for deriving KAM-type linearization theorems directly -- and almost immediately -- from the existence of the stable foliation for a renormalization operator. We give a few illustrations in dynamics in one and several complex variables, starting with a version of the classical theorem of Arnol'd and ending with a result on persistence of Herman rings in families of two-dimensional maps.
The paper explores scaling properties of bubbles -- a complex analogue of Arnold tongues, associated to a one-dimensional family of analytic circle diffeomorphisms. Bubbles are smooth loops in the upper half-plane attached at all rational points of the real line. Results of a paper by X.~Buff and N.~Goncharuk (2015) show that the size of a $p/q$-bubble has order at most $q^{-2}$. In the current paper we improve this estimate by showing that the size of a $p/q$-bubble near a bounded-type irrational number $α$ has order $d^{ξ(α)} \cdot q^{-2}$, where $ξ(α)>0$, and $d$ is the distance between $α$ and $p/q$. Proofs are based on a renormalization technique. In particular, $ξ(α)$ is related to the unstable and the top stable eigenvalues of the renormalization operator at the rotation by $α$.
We prove that the classification of real-analytic vector fields on the two-torus up to orbital topological equivalence does not admit a complete numerical invariant that is a Borel function. Moreover, smooth vector fields that are difficult to classify appear in generic smooth 7-parameter families. In dimension 2, this improves the recent result of Gorodetski and Foreman (arXiv:2206.09322) for non-classifiability of smooth diffeomorphisms up to continuous conjugacy.
Authomorphic or $s$-measures for circle diffeomorphisms were introduced by R.Douady and J.-C. Yoccoz in 1999. They have multiple applications in circle dynamics, with the case $s=-1$ being particularly important for describing conjugacy classes. In arxiv:2306.13524, E. de Faria, P. Guarino and B. Nussenzveig proved existence and uniqueness of automorphic $s$-measures for multicritical circle maps for all $s>0$. The purpose of this paper is to extend these results to $s$-measures with negative values of $s$. As an application, we prove a smoothness result for irrational Arnold tongues in families of multicritical maps.
We study the global behavior of the renormalization operator on a specially constructed Banach manifold that has cubic critical circle maps on its boundary and circle diffeomorphisms in its interior. As an application, we prove results on smoothness of irrational Arnold tongues.
We consider holomorphic maps defined in an annulus around $\mathbb R/\mathbb Z$ in $\mathbb C/\mathbb Z$. E. Risler proved that in a generic analytic family of such maps $f_ζ$ that contains a Brjuno rotation $f_0(z)=z+α$, all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near $f_0$. In this paper, we obtain the Risler's result as a corollary of the following construction. We introduce a renormalization operator on the space of univalent maps in a neighborhood of $\mathbb R/\mathbb Z$. We prove that this operator is hyperbolic, with one unstable direction corresponding to translations. We further use a holomorphic motions argument and Yoccoz's theorem to show that its stable foliation consists of diffeomorphisms that are conjugate to rotations.
The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved that under mild conditions of the Diophantine type on the rotation number any two $C^{2+α}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-ν}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from the linearizable case of circle diffeomorphisms where conjugacy is arbitrarily smooth, or even analytic, for sufficiently smooth diffeomorphisms.
We study bifurcations in finite-parameter families of vector fields on $S^2$. Recent papers by Yu. Ilyashenko, N. Goncharuk, Yu. Kudryashov, I. Schurov, and N. Solodovnikov provide examples of (locally generic) structurally unstable families of 3-parameter vector fields: generic close 3-parameter families experience different bifurcations. In this paper, we use these results to construct new examples of few-parameter generic families of planar vector fields such that their classification has many invariants. In particular, we construct (a) 3-parameter families with infinitely many numerical invariants; (b) 4-parameter families with arbitrarily many "robust" numerical invariants; (c) 5-parameter families with functional invariants.
We study global bifurcations in generic 3-parameter families of vector fields on $S^2$. In the recent article [arXiv:1506.06797], Ilyashenko, Kudryashov, and Schurov show that 3-parameter unfoldings of vector fields with the polycycle "tears of the heart" are structurally unstable. We consider 3-parameter unfoldings of vector fields with separatrix graphs "ears" and "glasses", and prove that these families are structurally unstable as well. We also study in more details the classical bifurcation of a saddle loop, and use it as a building block in our main example.
We classify global bifurcations in generic one-parameter local families of \vfs on $S^2$ with a parabolic cycle. The classification is quite different from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable.
In the study of global bifurcations of vector fields on $S^2$, it is important to distinguish a set "where the bifurcation actually occurs", -- the bifurcation support. Hopefully, it is sufficient to study the bifurcation in a neighborhood of the support only. The first definition of bifurcation support was proposed by V.Arnold. However this set appears to be too small. In particular, the newly discovered effect, an open domain in the space of three-parametric families on $S^2$ with no structurally stable families, is not visible in a neighborhood of the bifurcation support. In this article, we give a new definition of "large bifurcation support" that accomplishes the task. Roughly speaking, if we know the topological type of the phase portrait of a vector field, and we also know the bifurcation in a neighborhood of the large bifurcation support, then we know the bifurcation on the whole sphere.
Bubbles is a fractal-like set related to a circle diffeomorphism; they are a complex analogue to Arnold tongues. In this article, we prove an approximate self-similarity of bubbles.
"Tears of the heart" is a hyperbolic polycycle formed by three separatrix connections of two saddles. It is met in generic 3-parameter families of planar vector fields. In [arXiv:1506.06797], it was discovered that generically, the bifurcation of a vector field with "tears of the heart" is structurally unstable. The authors proved that the classification of such bifurcations has a numerical invariant. In this article, we study the bifurcations of "tears of the heart" in more detail, and find out that the classification of such bifurcation may have arbitrarily many numerical invariants.
The construction of complex rotation numbers, due to V.Arnold, gives rise to a fractal-like set "bubbles" related to a circle diffeomorphism. "Bubbles" is a complex analogue to Arnold tongues. This article contains a survey of the known properties of bubbles, as well as a variety of open questions. In particular, we show that bubbles can intersect and self-intersect, and provide approximate pictures of bubbles for perturbations of Möbius circle diffeomorphisms.
In this article, we prove two results. First, we construct a dense subset in the space of polynomial foliations of degree $n$ such that each foliation from this subset has a leaf with at least $\frac{(n+1)(n+2)}2-4$ handles. Next, we prove that for a generic foliation invariant under the map $(x, y)\mapsto (x, -y)$ all leaves have infinitely many handles.
Consider a holomorphic foliation with singularities of a 2-dimensional complex manifold. In this article we prove a new sufficient condition for this foliation to have countably many homologically independent complex limit cycles. In particular, if all leaves of a foliation are dense in the phase space, and it has a complex hyperbolic singular point, then it has infinitely many homologically independent complex limit cycles.