The Schauder fixed point theorem
The Schauder fixed-point theorem for connectivity maps
The semidynamical system near a closed negatively strongly invariant set.
The sequence entropy for Morse shifts and some counterexamples
The set of recurrent points of a continuous self-map on compact metric spaces and strong chaos
We discuss the existence of an uncountable strongly chaotic set of a continuous self-map on a compact metric space. It is proved that if a continuous self-map on a compact metric space has a regular shift invariant set then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.
The simultaneous nonlinear inequalities problem and applications.
The singular Cauchy-Nicoletti problem for the system of two ordinary differential equations
In the paper the singular Cauchy-Nicoletti problem for the system ot two ordinary differential equations is considered. New sufficient conditions for solvability of this problem are proved. In the proofs the topological method is applied. Some comparisons with known results are also given in the paper.
The solenoids are the only circle-like continua that admit expansive homeomorphisms
A homeomorphism h:X → X of a compactum X is expansive provided that for some fixed c > 0 and any distinct x, y ∈ X there exists an integer n, dependent only on x and y, such that d(hⁿ(x),hⁿ(y)) > c. It is shown that if X is a circle-like continuum that admits an expansive homeomorphism, then X is homeomorphic to a solenoid.
The spectrum and commutant of a certain weighted translation operator.
The structure of orbits in dynamical systems
The structure of the -ideal of -porous sets
We show a general method of construction of non--porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non--porous Suslin subset of a topologically complete metric space contains a non--porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non--porous element. Namely, if we denote the space of all compact subsets of a compact metric space with the Vietoris topology...
The structure of -limit sets for continuous maps of the interval
We prove that every infinite nowhere dense compact subset of the interval is an -limit set of homoclinic type for a continuous function from to .
The Subset P+ And P- Of The Split Interval
The subsets P+ and P- of the split interval.
The topological centralizers of Toeplitz flows and their Z2-extensions.
The topological centralizers of Toeplitz flows satisfying a condition (Sh) and their Z2-extensions are described. Such Toeplitz flows are topologically coalescent. If {q0, q1, ...} is a set of all except at least one prime numbers and I0, I1, ... are positive integers then the direct sum ⊕i=0∞ Zqi|i ⊕ Z can be the topological centralizer of a Toeplitz flow.
The topological complexity of sets of convex differentiable functions.
Let C(X) be the set of all convex and continuous functions on a separable infinite dimensional Banach space X, equipped with the topology of uniform convergence on bounded subsets of X. We show that the subset of all convex Fréchet-differentiable functions on X, and the subset of all (not necessarily equivalent) Fréchet-differentiable norms on X, reduce every coanalytic set, in particular they are not Borel-sets.
The topological degree and fixed points of DC-mappings
The Topological Entropy of the Transformation x ... ax (1-x).
The topological fixed point property - an elementary continuum-theoretic approach
A set contained in a topological space has the topological fixed point property if every continuous mapping of the set into itself leaves some point fixed. In 1969, R. H. Bing published his article The Elusive Fixed Point Property, posing twelve intriguing and difficult problems, which exerted a great influence on the study of the fixed point property. We now present a survey article intended for a broad audience that reports on this area of fixed point theory. The exposition is also intended to...
The topological Hawaiian earring group does not embed in the inverse limit of free groups.