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We prove three formulas for computing topological pressure of $C^1$-generic conservative diffeomorphism and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that…

For self-similar sets, there are two important separation properties: the open set condition and the weak separation condition introduced by Zerner, which may be replaced by the formally stronger finite type property of Ngai and Wang. We…

We show that Bowen's equation, which characterises the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In…

Let $\mu$ be a planar self-similar measure with similarity dimension exceeding $1$, satisfying a mild separation condition, and such that the fixed points of the associated similitudes do not share a common line. Then, we prove that the…

For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for…

We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a…

Let $(X,d,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $\mu $ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite…

We introduce the property of countable separation for a locally convex Hausdorff space $X$ and relate it to the existence of a metrizable coarser topology. Building on this, we demonstrate how the separability of $X$ is equivalent to the…

We consider abstract second order systems of the form $\ddot{x}(t) + D \dot{x}(t) + Sx(t)=0$, which are typically analyzed via the operator matrix $\mathcal{A}=\left[\begin{smallmatrix} 0 & I \\ -S & -D \end{smallmatrix}\right]$ governing…

The interplay between unitary dynamics and quantum measurements induces diverse phenomena in open quantum systems with no counterparts in closed quantum systems at equilibrium. Here, we generally classify Kraus operators and their effective…

For any 3-manifold M and any nonnegative integer g, we give here examples of metrics on M each of which has a sequence of embedded minimal surfaces of genus g and without Morse index bounds. On any spherical space form S^3/Gamma we…

Depending on a natural parameter $l$, we study the topological, metric, and fractal properties of the homogeneous self-similar set $$K_{l}=\left\{\sum_{i=1}^{\infty} \frac{\varepsilon_i}{(2l+2)^i} : (\varepsilon_i) \in \{0, 2, 4, \dots, 2l,…

The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H {x:\lim_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha}. \] For self-similar measures under the open set condition the behavior of…

We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on 3-manifold is completely defined by an em- bedding of two-dimensional stable and unstable heteroclinic…

The class of stochastically self-similar sets contains many famous examples of random sets, e.g. Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the…

The paper is devoted to the study of topological properties, structure and classification of Morse flows with fixed points on the boundary of three-dimensional manifolds. We construct a complete topological invariant of a Morse flow,…

Let $\{x\_n\}\_{n\geq 0}$ be a sequence of $[0,1]^d$, $\{\lambda\_n\} \_{n\geq 0}$ a sequence of positive real numbers converging to 0, and $\delta>1$. Let $\mu$ be a positive Borel measure on $[0,1]^d$, $\rho\in (0,1]$ and $\alpha>0$.…

Suppose M is a noncompact connected 2-manifold and m is a good Radon measure of M with m(partial M) = 0. Let H(M)_0 denote the identity component of the group of homeomorphisms of M equipped with the compact-open topology and let H(M; m)_0…

We study the set S of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation. Equivalently, the set S is formed by ergodic probability measures…

We obtain a series of results in the global theory of free boundary minimal surfaces, which in particular provide a rather complete picture for the way different complexity criteria, such as area, topology and Morse index compare, beyond…