Let X be a compact metric space and let f : X → X be continuous. It is well known that a given discrete dynamical system (X, f) induces a map Φ on the space F(X) of fuzzy sets. In this paper we study relations between sizes of topological entropies of a given crisp dynamical system (X, f) and its fuzzy counterpart (F(X),Φ). Among another things, we present a constructive proof of the fact that ...

Normal superconductors with Rashba spin-orbit coupling have been explored as candidate systems of topological superconductors. Here we present a comparative theoretical study of the effects of different types of disorder on the topological phases of two-dimensional Rashba spin-orbit coupled superconductors. First, we show that a topologically trivial superconductor can be driven into a chiral t...

What are the possible topological forms for a conformal dynamical system? Part of the answer is provided by two theorems, due to Thurston, which employ iteration on Teichmüller space to construct rational maps and Kleinian groups of a given topological form. More precisely, the iteration either finds a geometric model or reveals a topological obstruction to its existence. This dichotomy stems f...

We demonstrate experimentally the generation of square and hexagonal lattices of optical vortices and reveal their propagation in a saturable nonlinear medium. If the topological charges of the vortices have identical signs, the lattice exhibits rotation, whereas if their signs alternate between being the same and being opposite to each other, we observe stable propagation of the structures. In...

In this paper we propose a methodology for deriving a model of a complex system by exploiting the information extracted from Topological Data Analysis. Central to our approach is the S[B] paradigm in which a complex system is represented by a two-level model. One level, the structural S one, is derived using the newly introduced quantitative concept of Persistent Entropy. The other level, the b...

Let X1, X2 be mixing connected algebraic dynamical systems with the descending chain condition. We show that every equivariant continuous map X1 → X2 is affine (that is, X2 is topologically rigid) if and only if the system X2 has finite topological entropy.

Let (X,T ) be a topological dynamical system. We define the measuretheoretical lower and upper entropies hμ(T ), hμ(T ) for any μ ∈ M(X), where M(X) denotes the collection of all Borel probability measures on X. For any non-empty compact subset K of X, we show that htop(T,K) = sup{hμ(T ) : μ ∈M(X), μ(K) = 1}, htop(T,K) = sup{hμ(T ) : μ ∈M(X), μ(K) = 1}, where htop(T,K) denotes Bowen’s topologic...

The (ǫ, n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance ǫ during the time interval n. Behavior of the (ǫ, n)-complexity functions as n → ∞ is reflected in the properties of special measures. These measures are constructed as limits of atomic measures supported at...

We introduce a new topological semantics for belief logics in which the belief modality is interpreted as the interior of the closure of the interior operator. We show that the system wKD45, a weakened version of KD45, is sound and complete w.r.t. the class of all topological spaces. Moreover, we point out a problem regarding updates on extremally disconnected spaces that appears in the setting...