Dear Colleagues,

In recent decades, the topological theory of dynamical systems has rapidly developed. In particular, topological dynamical systems are nowadays widely applied in chaos theory, combinatorics, fractal geometry, etc. The theory of topological dynamical systems is one of the most interesting research fields in contemporary mathematics. Many of its theoretical results have found many real-world applications.

In the last twenty years, considerable attention has been paid to the theory’s connections with fractal sets and function spaces, especially towards links with wavelet analysis. Therefore, topological dynamical systems act as links between several mathematical fields, and in general between different fields of the functional analysis. In particular, topological dynamical systems deal with the topological properties of dynamical systems, i.e., the study of phenomena related to iterations of continuous maps defined on topological spaces. The topological approach to dynamical systems, due to the pioneering work of Henry Poincaré on the topological properties of differential equations, is relevant both in the qualitative theory of dynamical systems and in the numerical theory of dynamical systems. More importantly, dynamical systems may often possess different kinds of symmetry. Both continuous dynamical systems and discrete dynamical systems exhibit, in different ways, a local form of symmetry at minimum (e.g., Hénon map, Lorenz attractor). In fact, in order to understand the models that describe the world around us, we need to know the best way to model the symmetry of nature.

In this Special Issue, we invite and welcome review, expository, and original papers dealing with recent advances in the modern theory of topological dynamical systems, and, from a more general point of view, all theoretical and practical studies in pure and applied mathematics focused on this topic.

The main topics of this Special Issue include (but are not limited to):

1. Dynamical systems for fixed point problems and variational inequalities;
2. Operators, dynamical systems and global convergence in measurable function spaces;
3. Continuous dynamical systems and integral equations;
4. Discrete dynamical systems, iterative process and limit sets;
5. General theory of dynamical systems;
6. Orbits, attractors and iterated function systems;
7. Chaos theory, combinatorics and fractal sets;
8. Isomorphism of dynamical systems and integral transformations;
9. Random dynamical systems and applications.

Dr. Emanuel Guariglia
Guest Editor

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• dynamical systems
• global convergence
• integral equations
• orbits
• attractors
• fractal sets
• symmetry

Title: a new F-connected space(tentative title)
Authors: Elvis Aponte; Jhixon Macias; Luis Mejias; Jorge Vielma
Affiliation: Departamento de Matematicas, Facultad de Ciencias Naturales y Matem ´ aticas, Escuela Superior ´ Politecnica del Litoral (ESPOL), Campus Gustavo Galindo km. 30.5 V ´ ´ıa Perimetral, Guayaquil EC090112, Ecuador.
Abstract: We introduce a non-trivial strongly connected topology on the set N of the positive integers which arises some interesting results. Among those, we have that such topology is F-connected, in other words, it is both hyperconnected and ultraconnected. From this, we deduce that some subsets of N are F-connected too. Futhermore, we prove that there is a non-trivial maximal hyperconnected topology on the prime numbers joint with 1, which is not T1.