arXiv subjects

Nina Snigireva

Publications and source records attributed to Nina Snigireva.

8 recordsLinked to original sources

arXiv2024

Holomorphic functions on complex Banach lattices

We introduce and study the algebraic, analytic and lattice properties of regular homogeneous polynomials and holomorphic functions on complex Banach lattices. We show that the theory of power series with regular terms is closer to the theory of functions of several complex variables than the theory of holomorphic functions on Banach spaces. We extend the concept of the Bohr radius to Banach lattices and show that it provides us with a lower bound for the ratio between the radius of regular convergence and the radius of convergence of a regular holomorphic function. This allows us to show that in finite dimensions the radius of convergence of the Taylor series of a holomorphic function coincides with the radius of convergence of its monomial expansion but that on $\ell_p$ these two radii can be radically different.

math.FA↗

arXiv2022

Strongly-Fibred Iterated Function Systems and the Barnsley--Vince triangle

We review the theory of semiattractors associated with non-contractive Iterated Function Systems (IFSs) and demonstrate its applications on a concrete example. In particular, we present criteria for the existence of semiattractors due to Lasota and Myjak. We also discuss the Kieninger criterion which allows us to characterise when a semiattractor is strongly-fibred. Finally, we consider a specific example of a non-contractive IFS introduced by Barnsley and Vince. We find an invariant measure for this system which allows us to describe its semiattractor. The difficulty in analysing this IFS stems from the fact that it is neither eventually contractive nor contractive on average.

math.DS↗

arXiv2022

Transition Phenomena for the Attractor of an Iterated Function System

Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Recently, however, there has been an interest in what occurs to the attractor at the boundary between contractvity and expansion of the IFS. That is the subject of this paper. For a family $F_t$ of IFSs depending on a real parameter $t>0$, the existence and properties of two types of transition attractors, called the lower transition attractor $A_{\bullet}$ and the upper transition attractor $A^{\bullet}$, are investigated. A main theorem states that, for a wide class of IFS families, there is a threshold $t_0$ such that the IFS $F_t$ has a unique attractor $A_t$ for $t t_0$. At the threshold $t_0$, there is an $F_{t_0}$-invariant set $A^{\bullet}$ such that $A^{\bullet} = \lim_{t\rightarrow t_0} A_t$.

math.DS↗

arXiv2021

Orthogonally Additive Sums of Powers of Linear Functionals

Let $E$ be a Banach lattice, $λ_1,λ_2,\ldots,λ_k$ non-zero scalars and $φ_1,φ_2,\ldots,φ_k$ pairwise independent linear functionals on $E$. We show that if $k<m$ then $\sum_{j=1}^kλ_jφ_j^m$ is orthogonally additive if and only if $φ_j$ or $-φ_j$ is a lattice homomorphism for each $j$, $1\le j\le k$. Moreover, for each $m\ge 2$, we provide an example to show that this result does not extend to the case where $k=m$.

math.FA↗

arXiv2021

A Nakano Carrier Theorem for Polynomials

We use a localisation technique to study orthogonally additive polynomials on Banach lattices. We derive alternative characterisations for orthogonal additivity of polynomials and orthosymmetry of $m$-linear mappings. We prove that an orthogonally additive polynomial which is order continuous at one point is order continuous at every point and we give an example to show that this result does not extend to regular polynomials in general. Finally, we prove a Nakano Carrier theorem for orthogonally additive polynomials, generalising a result of Kusraev.

math.FA↗

arXiv2020

Weakly contractive iterated function systems and beyond: A manual

We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity of the random iteration algorithm ("chaos game"), can be obtained rather easily for weakly contractive systems. We show that the class of attractors of weakly contractive IFSs is essentially wider than the class of classical IFSs' fractals. On the other hand, we show that, in reasonable spaces, a typical compact set is not an attractor of any weakly contractive IFS. We explore the possibilities and restrictions to break the contractivity barrier by employing several tools from fixed point theory: geometry of balls, average contractions, remetrization technique, ordered sets, and measures of noncompactness. From these considerations it follows that while the existence of invariant sets and invariant measures can be assured rather easily for general iterated function systems under mild conditions, to establish the existence of attractors and unique invariant measures is a substantially more difficult problem. This explains the central role of contractive systems in the theory of IFSs.

math.DS↗

arXiv2018

Geometry of Spaces of Orthogonally Additive Polynomials on C(K)

We study the space of orthogonally additive $n$-homogeneous polynomials on $C(K)$. There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive $n$-homogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach-Stone theorem. We conclude with a classification of the exposed points.

math.FA↗

arXiv2017

Embedding the symbolic dynamics of Lorenz maps

Necessary and sufficient conditions for the symbolic dynamics of a Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

math.DS↗