We study the skew product systems $T: \mathbb{S}^1\times\mathbb{R}^d \to \mathbb{S}^1 \times\mathbb{R}^d$, \begin{equation*} T(x,y)=(\ell x, A y+\phi(x)), \end{equation*} where $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$ with $\rho(A)<1$, and $\phi \in C^r(\mathbb{S}^1, \mathbb{R}^d)$. We allow the fibers to have any dimension and $A$ to be non-conformal. We show that: when $|\det(A)|\ell<1$, for almost every $\phi$, the Hausdorff dimension of the solenoid attractor and the SRB measure equals the affinity dimension; when $|\det(A)|\ell>1$, for almost every $\phi$, the SRB measure is absolutely continuous. We introduce a derivative dispersion condition, replacing the usual transversality condition.
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that $\v...
Let $n \geq 4$ and $\rho: \pi_1S \rightarrow PSL(n,\mathbb{R})$ be a Hitchin representation. We study the topology of a cocompact domain of discontinuity $\Omega_{\rho}$ in the flag manifold $\mathcal{F}_{1,n-1}$ of line-hyperplane pairs in $\mathbb{R}^n$ defined by Guichard-Wienhard. In particular, we lift $\Omega$ to...
Let $g:\mathbb{D}\to\mathbb{D}$ be a hyperbolic holomorphic self-map of the unit disk with Denjoy--Wolff point $\tau\in\partial\mathbb{D}$ and angular derivative $\alpha\in(0,1)$. We say that $g$ has an \textit{extremal hyperbolic rate} if its forward iterates satisfy the sharp metric asymptotic \begin{equation*} \rho_...
Let $\mathcal{U} \Subset \mathbb{C}$ be fixed, $\Phi(s)=e^s-s-1$, $F(\nu)=\frac{\nu}{2+|\nu|}$, and let $f^{F(\nu)}$ denote the principal solution of the corresponding Beltrami equation. The identity $K_{F(\nu)}=1+|\nu|$ identifies compactly supported David coefficients with exponential-Orlicz parameters. We prove that...
We consider the parametrised family of sets $$ E(x,y)=\left\{\sum_{n=1}^{\infty}\frac{\varepsilon_n}{4^n}: (\varepsilon_n) \in \{0,x,y,x+y\}^{\mathbb{N}}\right\} $$ for $(x,y) \in \mathbb{N}^2$. This family can be viewed through three lenses: (a) as homogeneous self-similar sets; (b) as achievement sets of bi-geometric...
Natalia Jurga, D. Karvatskyi· 0 citations
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