Nina Snigireva

Let A be the attractor of a Banach contractive iterated function system (IFS) on a complete space. We prove that the orbit generated by a typical (in the sense of Baire category) driver recovers A with every possible speed. Our result extends the one from the paper: Leśniak et al. (Chaos 32(1):013110, 2022). We also show that our result is optimal from a certain point of view.

Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. Moreover, 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. The converse question, does the existence of an attractor imply that the IFS is contractive, originates in a 1959 work by Bessaga which proves a converse to the contraction mapping theorem. Although a converse is true in that case, it is known that it does not always hold for an IFS. In general, there do exist IFSs with attractors and which are not contractive. However, in the context of IFSs in Euclidean space, this question has been open. In this paper we show that a highly non-contractive iterated function system in Euclidean space can have an attractor. In order to do that, we introduce the concept of an L-expansive map, i.e., a map that has Lipschitz constant strictly greater than one under any remetrization. This is necessitated by the absence of positively expansive maps on the interval.

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 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 [Orthosymmetric bilinear operators, Vladikavkaz, preprint, 2007].

Iterated function systems (IFSs) and their attractors have been central in fractal geometry. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Two natural questions concerning contractivity arise. First, whether an IFS needs to be contractive to admit an attractor? Second, what occurs to the attractor at the boundary between contractivity and expansion of an IFS? The first question is addressed in the paper by providing examples of highly noncontractive IFSs with attractors. The second question leads to the study of two types of transition phenomena associated with an IFS family that depend on a real parameter. These are called lower and upper transition attractors. Their existence and properties are the main topic of this paper. Lower transition attractors are related to the semiattractors, introduced by Lasota and Myjak in 1990s. Upper transition attractors are related to the problem of continuous dependence of an attractor upon the IFS. A main result states that, for a wide class of IFS families, there is a threshold such that the IFSs in the one-parameter family have an attractor for parameters below the threshold and they have no attractor for parameters above the threshold. At the threshold there exists a unique upper transition attractor.

The rate of convergence of the chaos game algorithm for recovering attractors of contractive iterated function systems (IFSs) is studied. As with successive Picard iterates in the Banach fixed point principle, one has the exponential convergence. However, a symbolic sequence driving the iteration needs to obey some suitable statistical properties. Specifically, this sequence needs to behave like the classical Champernowne sequence. The exponent of convergence can be estimated from below in terms of (lower and upper) box dimensions of the attractor and from above by the entropy of the driver discounted by the Lipschitz constant of the IFS. Generically (in the sense of the Baire category), a driver that recovers the attractor yields arbitrarily slow convergence (of infinite order) interlaced with arbitrarily fast possible convergence (of order approaching a lower dimension).

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.