Related papers: Universal holomorphic maps with slow growth II. fu…

We investigate the hypercyclic properties of commutator maps acting on separable ideals of operators. As the main result we prove the commutator map induced by scalar multiples of the backward shift operator fails to be hypercyclic on the…

Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…

For any countable group, and also for any locally compact second countable, compactly generated topological group, G, we show the existence of a "universal" hypercyclic (i.e. topologically transitive) representation on a Hilbert space, in…

For $n\ge 3$, let $\Omega$ be a bounded domain in $R^n$ and $N$ be a compact Riemannian manifold in $R^L$ without boundary. Suppose that $u_n\in W^{1,n}(\Omega,N)$ are the Palais-Smale sequences of the Dirichlet $n$-energy functional and…

We provide criteria for the existence of upper frequently hypercyclic subspaces and for common hypercyclic subspaces, which include the following consequences. There exist frequently hypercyclic operators with upper-frequently hypercyclic…

Self-Organizing Maps (SOMs, Kohonen networks) belong to neural network models of the unsupervised class. In this paper, we present the generalized setup for non-Euclidean SOMs. Most data analysts take it for granted to use some subregions…

We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of…

Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds $X$ of dimension $<n$ to the complement $\mathbb{C}^n\setminus L$ of a compact convex set $L\subset\mathbb{C}^n$ satisfy the basic Oka property with approximation…

In this note, we consider the space $H(\Omega)^{\mathbb N}$ of sequences of holomorphic functions on an open set $\Omega\subset {\mathbb C}$. If $H(\Omega)$ is endowed with its natural topology and $H(\Omega)^{\mathbb N}$ is endowed with…

Let $\Sigma$ be a compact oriented surface and $N$ a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. For a solution of harmonic map flow starting from an almost-holomorphic map $\Sigma \to N$ (in the energy…

Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…

We establish rigidity results for holomorphic mappings and plurisubharmonic functions in complex geometry. First, under mild conditions, we show that the gradient of a $\operatorname{U}(1)$-invariant strictly plurisubharmonic function in…

We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic skew-products with a non-uniformly expanding base map. The aimed transformation preserves a foliation which is almost everywhere uniformly…

Let $\MSpace$ be a locally compact metric space and let $\pMap:\MSpace\to\MSpace$ be a not expansive map. We prove that for each $\ppa_0\in\MSpace$ the sequence $\ppa_0,\pMap(\ppa_0),\pMap^2(\ppa_0),\ldots$ is either relatively compact in…

We consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and we establish sharp uniform lower bounds of this invariant for the following classes of symplectomorphisms of…

We introduce new vanishing subspaces of the homogeneous H\"{o}lder space $\dot{C}^{0,\omega}(X,Y)$ in the generality of a doubling modulus $\omega$ and normed spaces $X$ and $Y.$ For many couples $X,Y,$ we show these vanishing subspaces to…

Let $D$ be the open unit disc in the complex plane. We denote by $\mathbb{C}$ the set of complex numbers and consider any compact set $K$ which is disjoint from $D$ and which also has connected complement. Let $A(K)$ denote all the…

We revisit the problem of characterizing cyclic elements for the shift operator in a broad class of radial growth spaces of holomorphic functions on the unit disk, focusing on functions of finite Nevanlinna characteristic. We provide…

We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the…

As a higher dimensional version of the theory of Morse functions, there have been various studies of smooth manifolds using generic smooth maps. As fundamental results, in these studies, they have found that inverse images of such maps…