Hyperbolic endomorphisms and overlap numbers

Hyperbolic endomorphisms and overlap numbers for equilibrium measures are studied on lifts of invariant sets. We look into the structure of Rokhlin conditional measures with respect to various fiber partitions, and find relations between them. We also compute topological overlap numbers in several
concrete cases. In particular, we obtain a large class of endomorphisms which asymptotically are irrational-to-1.

Some aspects of topological Galois theory

We establish a number of results on the subject of the first author's
topos-theoretic generalization of Grothendieck's Galois formalism. In
particular, we generalize in this context the existence theorem of algebraic closures, we give a concrete description of the atomic completion
of a small category whose opposite satisfies the amalgamation pro-
perty, and we explore to which extent a model of a Galois-type theory
is determined by its symmetries.

Three Hopf algebras from number theory, physics and topology, and their common operadic, simplicial and categorical background

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology.
The primary examples are the Hopf algebras of Goncharov for multiple zeta values, that of Connes--Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces.
We show that these examples can be successively unified by considering simplicial objects,
cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.

On the Ramanujan conjecture for automorphic forms over function fields I. Geometry

Global bifurcations of limit cycles in the Leslie-Gover model with the Allee effect

In this paper, we complete the global qualitative analysis of the Leslie-Gover system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations, we prove that such a system can have at most two limit cycles surrounding one singular point. We also conduct some numerical experiments to illustrate the obtained results.

Multi-parameter planar polynomial dynamical systems

In this paper, we study multi-parameter planar dynamical systems and carry out the global bifurcation analysis of such systems. To control the global bifurcations of limit cycle in these systems, it is necessary to know the properties and combine the effects of all their field rotation parameters. It can be done by means of the development of our bifurcational geometric method based on the Wintner-Perko termination principle and application of canonical
systems with field rotation parameters. Using this method, we solve, e.g., Hilbert's Sixteenth Problem on the maximum number of limit cycles and their distribution for the general Li\'{e}nard
polynomial system and a Holling-type quartic dynamical system. We also conduct some numerical experiments to illustrate the obtained results.

Pre-Calabi-Yau algebras as noncommutative Poisson structures

We show how the double Poisson algebra introduced in \cite{VdB} appear as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on $Ac part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structure. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson bracket on representation spaces $({\rm Rep}_n A)^{Gl_n}$ for any associative algebra $A$.

Introduction to Higher Cubical Operads. Second Part: The Functor of Fundamental Cubical Weak $\infty$-Groupoids for Spaces

Some Aspects of Dynamical Topology: Dynamical Compactness and Slovak Spaces

The area of dynamical systems where one investigates
dynamical properties that can be described in topological terms is "Topological Dynamics".
Investigating the topological
properties of spaces and maps that can be described in dynamical
terms is in a sense the opposite idea. This area is recently called as "Dynamical
Topology". For (discrete) dynamical systems given by compact metric spaces and
continuous (surjective) self-maps we (mostly) survey some results on two
new notions: "Slovak Space" and "Dynamical Compactness". Slovak space
is a dynamical analogue of the rigid space: a nontrivial compact
metric space whose homeomorphism group is cyclic and generated by a
minimal homeomorphism.
Dynamical compactness is a new concept of chaotic dynamics. The
omega-limit set of a point is a basic notion in theory of dynamical
systems and means the collection of states which "attract" this point
while going forward in time. It is always nonempty when the phase
space is compact. By changing the time we introduced the notion of the
omega-limit set of a point with respect to a Furstenberg family. A
dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness:
all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.

From Euler's play with infinite series to the anomalous magnetic moment