Related papers: Local structure of self-affine sets
We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We…
Floyd gave an example of a minimal dynamical system which was an extension of an odometer and the fibres of the associated factor map were either singletons or intervals. Gjerde and Johansen showed that the odometer could be replaced by any…
Non-autonomous self-similar sets are a family of compact sets which are, in some sense, highly homogeneous in space but highly inhomogeneous in scale. The main purpose of this note is to clarify various regularity properties and separation…
We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit,…
We prove that a self-similar Cantor set in $\mathbb{Z}_N \times \mathbb{Z}_N$ has a fractal uncertainty principle if and only if it does not contain a pair of orthogonal lines. The key ingredient in our proof is a quantitative form of…
In the realm of Delone sets in locally compact, second countable, Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on…
The complement of a Cantor set in the complex plane is itself regarded as a Riemann surface of infinite type. The problem is the quasiconformal equivalence of such Riemann surfaces. Particularly, we are interested in Riemann surfaces given…
In 1994, J.Cobb constructed a tame Cantor set in $\mathbb R^3$ each of whose projections into $2$-planes is one-dimensional. We show that an Antoine's necklace can serve as an example of a Cantor set all of whose projections are…
We characterize analytic curves that contain non-trivial self-affine sets. We also prove that compact algebraic surfaces cannot contain non-trivial self-affine sets.
In this paper, we consider the connectedness of planar self-affine set $T(A,\mathcal{D})$ arising from an integral expanding matrix $A$ with characteristic polynomial $f(x)=x^2+bx+c$ and a digit set $\mathcal{D}=\{0,1,\dots, m\}v$. The…
We investigate homeomorphisms of a compact interval, applied randomly. We consider this system as a skew product with the two-sided Bernoulli shift in the base. If on the open interval there is a metric in which almost all maps are…
Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an…
Self-similar sets with open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples…
Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to…
Nonexpansive mappings play a central role in modern optimization and monotone operator theory because their fixed points can describe solutions to optimization or critical point problems. It is known that when the mappings are sufficiently…
In the paper we discuss the algebraic structure of topological full group $[[T]]$ of a Cantor minimal system $(X,T)$. We show that the topological full group $[[T]]$ has the structure similar to a union of permutational wreath products of…
We calculate the Assouad dimension of a planar self-affine set $X$ satisfying the strong separation condition and the projection condition and show that $X$ is minimal for the conformal Assouad dimension. Furthermore, we see that such a…
We consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomise the translation vectors with the stipulation that the column structure is…
We explore the occurrence of point configurations within non-meager (second category) Baire sets. A celebrated result of Steinhaus asserts that $A+B$ and $A-B$ contain an interval whenever $A$ and $B$ are sets of positive Lebesgue measure…
Labourie and the author independently showed that a convex real projective structure on an oriented surface of genus at least 2 is equivalent to a conformal structure plus a holomorphic cubic differential U. We analyze the behavior of the…