Related papers: Self-affine 2-attractors and tiles
Let $M$ be a $d\times d$ contracting matrix. In this paper we consider the self-affine iterated function system $\{Mv-u, Mv+u\}$, where $u$ is a cyclic vector. Our main result is as follows: if $|\det M|\ge 2^{-1/d}$, then the attractor…
A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation…
We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain…
We present new examples of open sets of diffeomorphisms such that a generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that…
A central question in the field of inhomogeneous attractors has been to relate the dimension of an inhomogeneous attractor to the condensation set and associated homogeneous attractor. This has been achieved only in specific settings, with…
Motivated by the theory of bi-singular pseudodifferential operators, we introduce a two dimensional version of the Adler-Manin trace. Our construction is rather general in the sense that it involves a twist afforded by an algebra…
We characterize when a compact, invariant, asymptotically stable attractor on a locally compact Hausdorff space is a strong deformation retract of its domain of attraction.
In this paper, we study heterodimensional cycles of two-parameter families of 3-dimensional diffeomorphisms some element of which contains nondegenerate heterodimensional tangencies of the stable and unstable manifolds of two saddle points…
Abstract polytopes are combinatorial structures with distinctive geometric, algebraic, or topological characteristics, that generalize (the face lattice of) traditional polyhedra, polytopes or tessellations. Most research has focused on…
This work continues the study of the properties of finitely constrained groups of binary tree automorphisms in terms of their Hausdorff dimension. We prove that there are exactly $2^{2d-3}$ finitely constrained groups of binary tree…
An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension…
We consider digit systems $(A,\mathcal{D})$, where $ A \in \mathbb{Q}^{n\times n}$ is an expanding matrix and the digit set $\mathcal{D}$ is a suitable subset of $\mathbb{Q}^n$. To such a system, we associate a self-affine set $\mathcal{F}…
An integral self-affine tile is the solution of a set equation $\mathbf{A} \mathcal{T} = \bigcup_{d \in \mathcal{D}} (\mathcal{T} + d)$, where $\mathbf{A}$ is an $n \times n$ integer matrix and $\mathcal{D}$ is a finite subset of…
We consider attractor varieties arising in the construction of dyonic black holes in Calabi-Yau compactifications of IIB string theory. We show that the attractor varieties are constructed from products of elliptic curves with complex…
In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study…
A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich…
A theory of single-element extensions of integer polymatroids analogous to that of matroids is developed. We present an algorithm to generate a catalog of $2$-polymatroids, up to isomorphism. When we implemented this algorithm on a…
Using Conley theory we show that local attractors remain (past) attractors under small non-autonomous perturbations. In particular, the attractors of the perturbed systems will have positive invariant neighborhoods and converge upper…
We study the bifurcation diagram of the R\"ossler system. It displays the various dynamical regimes of the system (stable or chaotic) when a parameter is varied. We choose a diagram that exhibits coexisting attractors and banded chaos. We…
The article focuses on word (or string) attractors, which are sets of positions related to the text compression efficiency of the underlying word. The article presents two combinatorial algorithms based on Suffix automata or Directed…