arXiv subjects
Publications and source records attributed to Igors Gorbovickis.
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.
Using degenerations of rational maps and local asymptotics of periodic multipliers, we prove that for every $d\ge 2$, the multipliers of any $2d-2$ distinct periodic orbits of degree $d$ rational maps are algebraically independent over $\mathbb C$, provided that at most $d$ of the selected orbits are fixed points. This condition is sharp because of the Holomorphic Index Formula that relates the multipliers of the $d+1$ fixed points. The result of this paper removes the additional restrictions on periods present in earlier work. The proof proceeds by induction on the degree, using a one-hole degeneration for the induction step and a three-hole degeneration in an exceptional degree four case.
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 $α$.
Given a number field $\mathbb{K} \subset \mathbb{C}$ that is not contained in $\mathbb{R}$, we prove the existence of a dense set of entire maps $f \colon \mathbb{C} \rightarrow \mathbb{C}$ whose preperiodic points and multipliers all lie in $\mathbb{K}$. This contrasts with the case of rational maps. In addition, we show that there exists an escaping quadratic-like map that is not conjugate to an affine escaping quadratic-like map and whose multipliers all lie in $\mathbb{Q}$.
We establish the independence of multipliers for polynomial endomorphisms of $\mathbb C^n$ and endomorphisms of $\mathbb P^n.$ This allows us to extend results about the bifurcation measure and the critical height obtained in \cite{arXiv:2305.02246} to the case of polynomial endomorphisms of $\mathbb C^n$ for $n\geq 3$. An important step in the proof is the irreducibility of the spaces of endomorphisms with $N$ marked periodic points, which is of independent interest.
We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial $p_{\,\mathrm{Feig}}(z)=z^2+c_{\,\mathrm{Feig}}$. In addition to that, we construct a piecewise constant function on $[-2,2]$ that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials $p_c(z) = z^2+c$ with $c \in [-2,2]$. Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial $p_c(z)=z^2+c$, is a $C^1$-smooth function of the real parameter $c$ on the interval $c\in(c_{\,\mathrm{Feig}},-3/4)$.
We prove that if two analytic multicritical circle maps with the same bounded type rotation number are topologically conjugate by a conjugacy which matches the critical points of the two maps while preserving the orders of their criticalities, then the conjugacy necessarily has $C^{1+α}$ regularity, where $α$ depends only on the bound on the type of the rotation number. We then extend this rigidity result to $C^3$-smooth bi-cubic circle maps.
A parameter $c_0\in\mathbb C$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. We study the accumulation set $\mathcal X$ of the critical points of the multipliers, as $n\to\infty$. This study complements the equidistribution result for the critical points of the multipliers that was previously obtained by the authors. In particular, in the current paper we prove that the accumulation set $\mathcal X$ is bounded, path connected and contains the Mandelbrot set as a proper subset. We also provide a necessary and sufficient condition for a parameter outside of the Mandelbrot set to be contained in the accumulation set $\mathcal X$ and show that this condition is satisfied for an open set of parameters. Our condition is similar in flavor to one of the conditions that define the Mandelbrot set. As an application, we get that the function that sends $c$ to the Hausdorff dimension of $f_c$, does not have critical points outside of the accumulation set $\mathcal X$.
We construct complex a-priori bounds for certain infinitely renormalizable Lorenz maps. As a corollary, we show that renormalization is a real-analytic operator on the corresponding space of Lorenz maps.
A parameter $c_0\in\mathbb C$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. We prove that all critical points of period $n$ multipliers equidistribute on the boundary of the Mandelbrot set, as $n\to\infty$.
The multiplier $λ_n$ of a periodic orbit of period $n$ can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c$. We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter $c$ vanishes). We use this algorithm to compute critical points of $λ_n$ up to period $n=10$.
The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these balls does not increase as well. We give new results about central sets of subsets of a Riemannian manifold and apply these results to prove new special cases of the Kneser-Poulsen conjecture in the two-dimensional sphere and the hyperbolic plane.
We construct a renormalization operator which acts on analytic circle maps whose critical exponent $α$ is not necessarily an odd integer $2n+1$, $n\in\mathbb N$. When $α=2n+1$, our definition generalizes cylinder renormalization of analytic critical circle maps. In the case when $α$ is close to an odd integer, we prove hyperbolicity of renormalization for maps of bounded type. We use it to prove universality and $C^{1+α}$-rigidity for such maps.
We define an analytic setting for renormalization of unimodal maps with an arbitrary critical exponent. We prove the global Hyperbolicity of Renormalization conjecture for unimodal maps of bounded type with a critical exponent which is sufficiently close to an even integer.
We consider the space of degree $n\ge 2$ rational maps of the Riemann sphere with $k$ distinct marked periodic orbits of given periods. First, we show that this space is irreducible. For $k=2n-2$ and with some mild restrictions on the periods of the marked periodic orbits, we show that the multipliers of these periodic orbits, considered as algebraic functions on the above mentioned space, are algebraically independent over $\mathbb C$. Equivalently, this means that at its generic point, the moduli space of degree $n$ rational maps can be locally parameterized by the multipliers of any $2n-2$ distinct periodic orbits, satisfying the above mentioned conditions on their periods. This work extends previous similar result (arXiv:1305.0867) obtained by the author for the case of complex polynomial maps.
The Kneser-Poulsen conjecture says that if a finite collection of balls in a d-dimensional Euclidean space is rearranged so that the distance between each pair of centers does not get smaller, then the volume of the union of these balls also does not get smaller. In this paper we prove that if in the initial configuration the intersection of any two balls has common points with no more than d+1 other balls, then the conjecture holds.
We present the following result: consider the space of complex polynomials of degree n>2 with n-1 distinct marked periodic orbits of given periods. Then this space is irreducible and the multipliers of the marked periodic orbits considered as algebraic functions on the above mentioned space, are algebraically independent over $\mathbb C$. Equivalently, this means that at its generic point, the moduli space of degree n polynomial maps can be locally parameterized by the multipliers of n-1 arbitrary distinct periodic orbits. A detailed proof of this result (together with a proof of a more general statement) is given in [arXiv:1305.0867]. In this exposition we substitute some of the technical lemmas from [arXiv:1305.0867] with more geometric arguments.