Related papers: Diffeomorphic approximation of piecewise affine ho…

Given an affine transformation $T$, we define its Fisher distortion $Dist_F(T)$. We show that the Fisher distortion has Riemannian metric structure and provide an algorithm for finding mean distorting transformation -- namely -- for a given…

Let $\Omega,\Omega'\subset\mathbb{R}^3$ be Lipschitz domains, let $f_m:\Omega\to\Omega'$ be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and $\sup_m \int_{\Omega}(|Df_m|^2+1/J^2_{f_m})<\infty$. Let $f$ be a weak…

The sequence of Dold coefficients $(a_n(f))$ of a self-map $f\colon X \to X$ forms a dual sequence to the sequence of Lefschetz numbers $(L(f^n))$ of iterations of $f$ under the M\"obius inversion formula. The set ${\mathcal AP}(f) = \{ n…

In an earlier article, we have "derived" space, as a part of the Random Dynamics project. In order to get locality we need to obtain reparametrization symmetry, or equivalently, diffeomorphism symmetry. There we sketched a procedure for how…

This paper has two themes that are intertwined: The first is the dynamics of certain piecewise affine maps on the Euclidean space that arise from a class of analog-to-digital conversion methods called Sigma-Delta quantization. The second is…

This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's…

Let S be a compact orientable surface and f be an element of the group Homeo_{0}(S) of homeomorphisms of S isotopic to the identity. Denote by F a lift of f to the universal cover of S. In this article, the following result is proved: if…

A natural quantity that measures how well a map $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is approximated by an affine transformation is…

A well-known open problem asks whether every bi-Lipschitz homeomorphism of $\mathbb{R}^d$ factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube $[0,1]^d$ into $\mathbb{R}^d$…

First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the…

In this article we study minimal homeomorphisms(all orbits are dense) of the tori $T^{n},$ $n<5.$ The linear part of a homeomorphism $\phi $ of $T^{n}$ is the linear mapping $L$ induced by $\phi $ on the first homology group of $T^{n}$. It…

A linear system of difference equations and a nonlinear perturbation are considered. We propose sufficient conditions to ensure that the homeomorphism of topological equivalence between them is actually a $C^1$ diffeomorphism. These…

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. The…

Let $\Omega$, $\Omega'\subset\mathbb{R}^n$ be bounded domains and let $f_m\colon\Omega\to\Omega'$ be a sequence of homeomorphisms with positive Jacobians $J_{f_m} >0$ a.e. and prescribed Dirichlet boundary data. Let all $f_m$ satisfy the…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…

We construct, for each irrational number $\alpha$, a minimal $C^1$-diffeomorphism of the circle with rotation number $\alpha$ which admits a measur

We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ of a Beltrami equation $\overline{\partial}f=\mu\,\partial f$ in a domain $D\subseteq\Bbb C$ is the so--called ring $Q-$homeomorphism with $Q(z)=K^T_{\mu}(z, z_0)$ where…

We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate…

The fundamental theorem of affine geometry says that a self-bijection $f$ of a finite-dimensional affine space over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then $f$ of the expected…

We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit $n$-dimensional cube onto itself, whose Jacobian is equal to $-1$ a.e. Moreover we prove that our homeomorphism can…