Related papers: A remark on uniform expansion

We introduce a new version of expansiveness for flows. Let $M$ be a compact Riemannian manifold without boundary and $X$ be a $C^1$ vector field on $M$ that generates a flow $\varphi_t$ on $M$. We call $X$ {\it rescaling expansive} on a…

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic…

We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of…

We provide a representation of the homomorphisms $U\longrightarrow \mathbb R$, where $U$ is the lattice of all uniformly continuous on the line. The resulting picture is sharp enough to describe the fine topological structure of the space…

We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant $\delta$. The bound is expressed in the uniform entropy integral of…

It is a well-known fact -- which can be shown by elementary calculus -- that the volume of the unit ball in $\mathbb{R}^n$ decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as $n \nearrow…

A {\it two-dimensional continued fraction expansion} is a map $\mu$ assigning to every $x \in\mathbb R^2\setminus\mathbb Q^2$ a sequence $\mu(x)=T_0,T_1,\dots$ of triangles $T_n$ with vertices $x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})\in\mathbb…

Let $G$ be a graph which satisfies $c^{-1} a^r \le |B(v,r)| \le c a^r$, for some constants $c,a>1$, every vertex $v$ and every radius $r$. We prove that this implies the isoperimetric inequality $|\partial A| \ge C |A| / \log(2+ |A|)$ for…

We generalize the dual notions of "expansion" and "collapse" so they can be applied to arbitrary metric spaces. We also expand the theory to allow for infinitely many such moves. Those tools are then employed to prove a variety of…

We relate the total curvature and the isoperimetric deficit of a curve $\gamma$ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $\gamma$. We provide also a Gauss-Bonnet theorem for a special class…

Let $M$ be a smooth, compact manifold and let $\mathcal{N}_{\mu}$ denote the set of Riemannian metrics on $M$ with smooth volume density $\mu$. For a given $g_0\in \mathcal{N}_{\mu}$, we show that if $\dim(M)\ge 5$, then there exists an…

We introduce the concept of topological expansive flow. We prove that this concept is invariant by topological conjugacy and reduces to expansivity in the compact case. We characterize tiopological expansive flows as rescaling expansive…

It is known that if a subset of $\mathbb{R}$ has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following…

We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent…

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only…

Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate…

We show that if there exists a topologically expansive homeomorphism on a uniform space, then the space is always a regular space. Through examples we show that in general composition of topologically expansive homeomorphisms need not be…

It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group $S_\infty$ is continuous. It is…

A compactness of the Revuz map is established in the sense that the locally uniform convergence of a sequence of positive continuous additive functionals is derived in terms of their smooth measures. To this end, we first introduce a metric…

Uniformly star superparacompactness, which is a topological property between compactness and completeness, can be characterized using finite-component covers and a measure of strong local compactness. Using these finite-component covers and…