The nonexistence of expansive homeomorphisms of dendroids
The nonexistence of universal metric flows
We consider dynamical systems of the form where is a compact metric space and is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract -limit sets, answering a question by Will Brian.
The omega limit sets of subsets in a metric space
In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the -limit set of is the limit point of the sequence in and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.
The ping-pong game, geometric entropy and expansiveness for group actions on Peano continua having free dendrites
It is shown that each expansive group action on a Peano continuum having a free dendrite must have a ping-pong game, and has positive geometric entropy when the acting group is finitely generated. As a corollary, it is shown that each Peano continuum having a free dendrite admits no expansive nilpotent group actions.
The real-analytic solutions of the Abel functional equation
For the Abel equation on a real-analytic manifold a dynamical criterion of solvability in real-analytic functions is proved.
The relationship between algebraic numbers and expansiveness of automorphisms on compact abelian groups
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 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 -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 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 Entropy of the Transformation x ... ax (1-x).
The topology of a moduli space for linear dynamical systems.
The universal minimal system for the group of homeomorphisms of the Cantor set
Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E) of self-homeomorphisms...
The variational principle
The Williams conjecture is false for irreducible subshifts.
The Williams conjecture is false for irreducible subshifts.