Related papers: Morse-Sard theorem for d.c. curves

We prove that if $E$ is a planar self-similar set with similarity dimension $d$ whose defining maps generate a dense set of rotations, then the $d$-dimensional Hausdorff measure of the orthogonal projection of $E$ onto any line is zero. We…

We prove that among the set of smooth diffeomorphisms there exists a $C^1$-open and dense subset of data such that either the Lagrange spectrum is finite and the dynamics is a Morse-Smale diffeomorphism or the Lagrange spectrum has positive…

In metric geometry, the question of whether a distance metric is given by the length of curves can be decided via the existence of midpoints with respect to the metric $d$. We adapt a similar characterization to the setting of Lorentzian…

We say that $f:[0,1]\to [0,1]$ is a {\it piecewise continuous interval map} if there exists a partition $0=x_0<x_1<\cdots<x_{d}<x_{d+1}=1$ of $[0,1]$ such that $f\vert_{(x_{i-1},x_i)}$ is continuous and the lateral limits $w_0^+=\lim_{x\to…

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves $\operatorname{Imm}(S^1,\mathbb{R}^d)$ and on its Sobolev…

We study the almost Mathieu operator at critical coupling. We prove that there exists a dense $G_\delta$ set of frequencies for which the spectrum is of zero Hausdorff dimension.

In the present paper we investigate the properties of the Hausdorff mapping $\mathcal{H}$, which takes each compact metric space to the space of its nonempty closed subspaces. It is shown that this mapping is nonexpanding (Lipschitz mapping…

This article considers isometries of the Kobayashi and Carath\'{e}od-ory metrics on domains in $ \mathbf{C}^n $ and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincar\'{e}'s theorem about…

We study the Hausdorff measure and dimension of the set of intrinsically simultaneously $\psi$-approximable points on a curve, surface, etc., given as a graph of integer valued polynomials. We obtain complete answers to these questions for…

Let $X$ be a partially ordered set with the property that each family of order intervals of the form $[a,b],[a,\rightarrow )$ with the finite intersection property has a nonempty intersection. We show that every directed subset of $X$ has a…

The Hausdorff dimension of the graphs of the functions in H\"older and Besov spaces (in this case with integrability p \geq 1) on fractal d-sets is studied. Denoting by s \in (0,1] the smoothness parameter, the sharp upper bound…

Our main result concerns the following condition: {\bf Condition C.} Let $X$ be a Banach space. A $C^1$ function $f:X\rightarrow \mathbb{R}$ satisfies Condition C if whenever $\{x_n\}$ weakly converges to $x$ and $\lim…

Consider a finite Blaschke product $f$ with $f(0) = 0$ which is not a rotation and denote by $f^n$ its $n$-th iterate. Given a sequence $\{a_n\}$ of complex numbers, consider the series $F(z) = \sum_n a_n f^n(z).$ We show that for any $w…

We obtain fractal Lipschitz-Killing curvature-direction measures for a large class of self-similar sets F in R^d. Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean…

We generalize the Arzel\`a-Ascoli theorem in the space of continuous maps on a compact interval with values in Euclidean N-space by providing a quantitative link between the Hausdorff measure of noncompactness in this space and a natural…

We prove that for any $C^r$ diffeomorphism, $f$, of a compact manifold of dimension $d>2$, $1\leq r\leq \infty$, admitting a transverse homoclinic intersection, we can find a $C^1$-open neighborhood of $f$ containing a $C^1$-open and…

In 1985 M. Smith constructed a nonmetric pseudo-arc; i.e. a Hausdorff homogeneous, hereditary equivalent and hereditary indecomposable continuum. Taking advantage of a decomposition theorem of W. Lewis, he obtained it as a long inverse…

In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented…

We study several notions of null sets on infinite-dimensional Carnot groups. We prove that a set is Aronszajn null if and only if it is null with respect to measures that are convolutions of absolutely continuous (CAC) measures on Carnot…

The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting…