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Karim Rakhimov

Publications and source records attributed to Karim Rakhimov.

12 recordsLinked to original sources
arXiv2026

Exact logarithmic Hausdorff and capacity exponents of the non-Brjuno set

Let $\mathcal{N}$ be the set of non-Brjuno real numbers. We determine the exact critical exponent of $\mathcal{N}$ for both logarithmic Hausdorff measure and logarithmic capacity. For the gauges $h_δ(r)=(\log(1/r))^{-δ}$, the critical exponent is $2$: the $h_δ$-Hausdorff measure is infinite locally for $0<δ\leq2$ and vanishes globally for $δ>2$, while the logarithmic capacity is positive exactly for orders $0<s\leq2$. In particular, $\dim_{\mathrm{cap}}\mathcal{N}=2$.

math.NT↗
arXiv2026

Linear response for random systems with a cusp

We study i.i.d.\ random compositions of cusp tent-like interval maps having a common cusp point and a common cusp value. For cusp exponents $-1<β<-\frac12$, we prove that the associated annealed transfer operators have a spectral gap on $W^{1,1}(I)$ and $W^{2,1}(I)$. For all sufficiently small perturbations of the probability law, there is a stationary density $h_\varepsilon\in W^{2,1}(I)$, unique among stationary densities belonging to $W^{1,1}(I)$. Moreover, the map $\varepsilon\mapsto h_\varepsilon$ is differentiable at $\varepsilon=0$ in $W^{1,1}(I)$, and we obtain an explicit linear response formula.

math.DS↗
arXiv2026

On the support of measures of large entropy for automorphisms of Kähler manifolds

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $X$ with simple action on cohomology. We show that every ergodic measure with sufficiently large entropy is supported on the Julia set of $f$. In particular, when $X$ is a surface, any ergodic measure with positive entropy is supported on the Julia set. The proof relies on quantitative estimates for the speed of convergence towards the Green currents of $f$, with respect to a suitable norm on an adapted functional space of non-necessarily closed currents.

math.DS↗
arXiv2025

Capacity dimension of the Brjuno set in $\mathbb{C}^n$

In this work, we prove that the complement of the Brjuno set in $\mathbb{C}^n$ has zero $C_σ$-capacity with respect to the kernel $k_σ(z,ξ)=\|z-ξ\|^{-2n+2}|\log{\|z-ξ\||^σ}$ for any $σ>n$. In particular, it follows that it has zero $h_δ$-Hausdorff measure with respect to the $h_δ(t)=t^{2n-2}|\log{t}|^{-δ}$, for any $δ>n+1$. This generalizes a previous result of Sadullaev and the second author in dimension one to higher dimensions.

math.CV↗
arXiv2024

On the support of measures of large entropy for polynomial-like maps

Let $f$ be a polynomial-like map with dominant topological degree $d_t\geq 2$ and let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the support of every ergodic measure whose measure-theoretic entropy is strictly larger than $\log \sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., the support of the unique measure of maximal entropy $μ$. The proof is based on the exponential speed of convergence of the measures $d_t^{-n}(f^n)^*δ_a$ towards $μ$, which is valid for a generic point $a$ and with a controlled error bound depending on $a$. Our proof also gives a new proof of the same statement in the setting of endomorphisms of $\mathbb P^k(\mathbb C)$ - a result due to de Thélin and Dinh - which does not rely on the existence of a Green current.

math.DS↗
arXiv2024

Flat structure of meromorphic connections on Riemann surfaces

The possible omega limit sets of simple geodesics for meromorphic connections on compact Riemann surfaces have been studied by Abate, Tovena and Bianchi. In this paper, we study the same problem for infinite self-intersecting geodesics. In the first part of the paper we study relation among meromorphic $k$-differentials, singular flat metrics and meromorphic connections. Moreover, we prove a Poincaré-Bendixson theorem for infinite self-intersecting geodesics of meromorphic connections with monodromy in $G$, where $\arg G^k=\{0\}$ for some $k\in\mathbb{N}$.

math.CV↗
arXiv2024

Hölder continuity and laminarity of the Green currents for Hénon-like maps

Under a natural assumption on the dynamical degrees, we prove that the Green currents associated to any Hénon-like map in any dimension have Hölder continuous super-potentials, i.e., give Hölder continuous linear functionals on suitable spaces of forms and currents. As a consequence, the unique measure of maximal entropy is the Monge-Ampère of a Hölder continuous plurisubharmonic function and has strictly positive Hausdorff dimension. Under the same assumptions, we also prove that the Green currents are woven. When they are of bidegree $(1,1)$, they are laminar. In particular, our results generalize results known until now only in algebraic settings, or in dimension 2.

math.CV↗
arXiv2024

Dynamics of Fuchsian meromorphic connections with real periods

In this paper, we study the dynamics of geodesics of Fuchsian meromorphic connections with real periods, giving a precise characterization of the possible $ω$-limit sets of simple geodesics in this case. The main tools are the study of the singular flat metric associated to the meromorphic connection, an explicit description of the geodesics nearby a Fuchsian pole with real residue larger than $-1$ and a far-reaching generalization to our case of the classical Teichmüller lemma for quadratic differentials.

math.CV↗
arXiv2023

Strong probabilistic stability in holomorphic families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$ and polynomial-like maps

We prove that, in stable families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, all invariant measures whose measure-theoretic entropy is strictly larger than $(k-1)\log d$ at a given parameter can be followed holomorphically with the parameter in all the parameter space. As a consequence, almost all points (with respect to any such measure at any parameter) in the Julia set can be followed holomorphically without intersections. This generalizes previous results by Berteloot, Dupont, and the first author for the measure of maximal entropy, and provides a parallel in this setting to the probabilistic stability of Hénon maps by Berger-Dujardin-Lyubich. Our proof relies both on techniques from the theory of stability/bifurcation in any dimension and on an explicit lower bound for the Lyapunov exponents for an ergodic measure in terms of its measure-theoretic entropy, due to de Thélin and Dupont. A local version of our result holds also for all measures supported on the Julia set with just strictly positive Lyapunov exponents and not charging the post-critical set. Analogous results hold in families of polynomial-like maps of large topological degree. In this case, as part of our proof, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure for a polynomial-like map in any dimension in term of its measure-theoretic entropy, generalizing to this setting the analogous result by de Thélin and Dupont valid on $\mathbb{P}^k(\mathbb{C})$.

math.DS↗
arXiv2023

Monotonicity of dynamical degrees for H{é}non-like and polynomial-like maps

We prove that, for every invertible horizontal-like map (i.e., H{é}non-like map) in any dimension, the sequence of the dynamical degrees is increasing until that of maximal value, which is the main dynamical degree, and decreasing after that. Similarly, for polynomial-like maps in any dimension, the sequence of dynamical degrees is increasing until the last one, which is the topological degree. This is the first time that such a property is proved outside of the algebraic setting. Our proof is based on the construction of a suitable deformation for positive closed currents, which relies on tools from pluripotential theory and the solution of the $d$, $\bar \partial$, and $dd^c$ equations on convex domains.

math.CV↗
arXiv2021

A mean value criterion for plurisubharmonic functions

In this paper we prove a criterion for plurisubharmonic functions in terms of integral mean by complex ellipsoids. Moreover, by using the criterion we prove an analogue of Blaschke-Privalov theorem for plurisubharmonic functions.

math.CV↗