Related papers: Global Topology from an Embedding

Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…

A topological classification of many classes of dynamical systems with regular dynamics in low dimensions is often reduced to combinatorial invariants. In dimension 3 combinatorial invariants are proved to be insufficient even for simplest…

I investigate a class of dynamical systems in which finite pieces of spacetime contain finite amounts of information. Most of the guiding principles for designing these systems are drawn from general relativity: the systems are…

Unsupervised feature learning often finds low-dimensional embeddings that capture the structure of complex data. For tasks for which prior expert topological knowledge is available, incorporating this into the learned representation may…

We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of…

As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth…

A system of N unidimensional global coupled maps (GCM), which support multiattractors is studied. We analize the phase diagram and some special features of the transitions (volume ratios and characteristic exponents), by controlling the…

Analyzing data from paleoclimate archives such as tree rings or lake sediments offers the opportunity of inferring information on past climate variability. Often, such data sets are univariate and a proper reconstruction of the system's…

We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of…

A few recent papers introduced the concept of topological synchronisation. We refer in particular to \cite{TS}, where the theory was illustrated by means of a skew product system, coupling two logistic maps. In this case, we show that the…

Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This…

Competition between two main attractors of the distributed chaos, one associated with translational symmetry (homogeneity) and another associated with rotational symmetry (isotropy), has been studied in freely decaying turbulence. It is…

This work explores the intersection of time-delay embeddings, periodic orbit theory, and symbolic dynamics. Time-delay embeddings have been effectively applied to chaotic time series data, offering a principled method to reconstruct…

It is shown that classical spaces with geometries emerge on boundaries of randomly connected tensor networks with appropriately chosen tensors in the thermodynamic limit. With variation of the tensors, the dimensions of the spaces can be…

Two geometric spaces are in the same topological class if they are related by certain geometric deformations. We propose machine learning methods that automate learning of topological invariance and apply it in the context of knot theory,…

The problem of learning a manifold structure on a dataset is framed in terms of a generative model, to which we use ideas behind autoencoders (namely adversarial/Wasserstein autoencoders) to fit deep neural networks. From a machine learning…

Let $M$ be a closed surface and $f$ a diffeomorphism of $M$. A diffeomorphism is said to permute a dense collection of domains, if the union of the domains are dense and the iterates of any one domain are mutually disjoint. In this note, we…

In this paper, we present a unified framework of multiple attractors including multistability, multiperiodicity and multichaos. Multichaos, which means that the chaotic solution of a system lies in different disjoint invariant sets with…

It is shown that in dimension at least three a local diffeomorphism of Euclidean n-space into itself is injective provided that the pull-back of every plane is a Riemannian submanifold which is conformal to a plane. Using a similar…

We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust…