The paper answers some questions asked by Sharkovski concerning the map F:(u,v) ↦ (u(4-u-v),uv) of the triangle Δ = u,v ≥ 0: u+v ≤ 4. We construct an absolutely continuous σ-finite invariant measure for F. We also prove the following strange phenomenon. The preimages of side I = Δ ∩ v=0 form a dense subset of Δ and there is another dense set Λ consisting of points whose orbits approach the interval I but are not attracted by I.
In this note we present the main ideas of the theory of the Conley index over a base space introduced in the papers [7, 8]. The theory arised as an attempt to solve two questions concerning the classical Conley index. In the definition of the index, the exit set of an isolating neighborhood is collapsed to a point. Some information is lost on this collapse. In particular, topological information about how a set sits in the phase space is lost. The first question addressed is how to retain some of...
In the present paper conditions are studied, under which a pseudo-orbit of a continuous map , where is a metric space, is shadowed, in a more general sense, by an accurate orbit of the map .
Absolute stability of a compact set is characterized by the cardinality of a fundamental system of positively invariant neighborhoods.
It is proved that any two composants of any two solenoids are homeomorphic.
Let F and G be finitely generated groups of polynomial growth with the degrees of polynomial growth d(F) and d(G) respectively. Let be a continuous action of F on a compact metric space X with a positive topological entropy h(S). Then (i) there are no expansive continuous actions of G on X commuting with S if d(G)
Defining the complexity of a green pattern exhibited by an interval map, we give the best bounds of the topological entropy of a pattern with a given complexity. Moreover, we show that the topological entropy attains its strict minimum on the set of patterns with fixed eccentricity m/n at a unimodal X-minimal case. Using a different method, the last result was independently proved in[11].
We consider a class of symbolic systems over a finite alphabet which are minimal almost one-to-one extensions of rotations of compact metric monothetic groups and provide computations of their enveloping semigroups that highlight their algebraic structure. We describe the set of idempotents of these semigroups and introduce a classification that can help distinguish between certain such systems having zero topological entropy.
Let G be a group generated by a set of affine unipotent transformations T: X → X of the form T(x) = A x + α, where A is a lower triangular unipotent matrix, α is a constant vector, and X is a finite-dimensional torus. We show that the enveloping semigroup E(X,G) of the dynamical system (X,G) is a nilpotent group and find upper and lower bounds on its nilpotency class. Also, we obtain a description of E(X,G) as a quotient space.