A𝕋-algebras from fiberwise essentially minimal zero-dimensional dynamical systems

Dynamical systems of finite-dimensional metric spaces and zero-dimensional covers

Let $\varphi$ be a homeomorphism on a totally disconnected, compact metric space $X$. We introduce a binary relation on the family of clopen subsets of $X$, which is described in terms of the $\varphi$-invariant probability measures. We show that $\varphi$ is uniquely ergodic if and only if any two clopen subsets of $X$ are comparable with respect to the binary relation.

A Baker omitted value, in short bov of a transcendental meromorphic function f is an omitted value such that there is a disk D centered at the bov for which each component of the boundary of \(f^{-1}(D)\) is bounded. Assuming all the iterates \(f^n\) to be analytic in a neighborhood of its bov, this article proves a number of results on the rotation domains of the function. The number of p-periodic Herman rings is shown to be finite for each \(p \ge 1\). It is also proved that every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that f has infinitely many repelling fixed points. If a repelling periodic point of period p is on the boundary of a p-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. As a corollary we have shown that if D is a p-periodic rotation domain of f and \(f^p\) is univalent in a neighbourhood of the boundary \( \partial D \) of D then f has no repelling p-periodic point on \( \partial D \). Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be a singleton. As a consequence, it is proved that if the boundary of a wandering domain W accumulates at some point of the plane under the iteration of f then each limit of \(f^n\) on W is either a parabolic periodic point or in the \(\omega \)-limit set of recurrent critical points. Using the same ideas, the boundaries of rotation domains are shown to be in the \(\omega \)-limit set of recurrent critical points.

(2008). On Ordering the Natural Numbers or The Sharkovski Theorem. The American Mathematical Monthly: Vol. 115, No. 2, pp. 159-165.

We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if that system is rigid. In the case that cocycles take values in the rotation group, it is also shown that rigidity implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately $K$-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps $C(X\times \T)$ onto $C(X\times \T)$.

The central problem in dynamical systems is the asymptotic behavior or topological structure of the orbits. Nevertheless only orbits of points with certain recurrence and form a set of full measure are truly of importance. Of course, such a set is desired to be as small (in the sense of set inclusion) as possible. In this paper we discuss such two sets: the set of weakly almost periodic points and the set of quasi-weakly almost periodic points. While the two sets are different from each other by definitions, we prove that their closures both coincide with the measure center (or the minimal center of attraction) of the dynamical systems. Generally, a point may have three levels of orbit-structure: the support of an invariant measure generated by the point, its minimal center of attraction and its ?-limit set. We study the three levels of orbit-structure for weakly almost periodic points and quasi-weakly almost periodic points. We prove that quasi-weakly almost periodic points possess especially rich topological orbit-structures. We also present a necessary and sufficient condition for a point to belong to its own minimal center of attraction.

We prove that a compact space embeds into a ‐product of compact metrizable spaces (‐product of intervals) if and only if is (strongly countable‐dimensional) hereditarily metalindelöf and every subspace of has a nonempty relative open second countable subset. This provides novel characterizations of ‐Corson and compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the ‐product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space , the property of being compact is determined by the topological structure of the space of continuous real‐valued functions of equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.

This paper discusses the bifurcation of periodic points of a generic symplectic diffeomorphism of a two manifold that depends on a parameter.A complete classification of the types of bifurcation that can occur in the generic case is given.Introduction.This paper discusses the bifurcation of periodic points of an area preserving diffeomorphism that depends on a parameter.In the spirit of Thorn and Smale only the generic case is considered.As in Smale's work on discrete dynamical systems the present problem was suggested by problems in ordinary differential equations (see [1]).Whereas Smale modeled his theory on differential equations with dissipation the present problem was suggested by conservative differential equations.Throughout this paper smooth will always mean C.Let M be a smooth, compact, two dimensional, symplectic manifold, S=SX the circle considered as a smooth manifold and F the space of all smooth mappings <p: MxS-> M with the property that for each s e S the map <ps=<p(-, s): M-> M is a symplectic diffeomorphism.Let F have the usual topology of C maps.The space F is complete and therefore is a Baire space.A point (x, s) e M x S is called a periodic point of least period m if <p(x)=x and m is the smallest positive integer for which this holds.Any multiple of m will be called a period of (x, s).If (x, s) is a periodic point of <p of least period m then each point (<p\ix), s) e M x S is a distinct periodic point of least period m for i = 1,..., m and the set U= i (<pl(x), s) is called the orbit of <p through (x, s).We wish to discuss the nature of the set of periodic points for <p e F and in particular the manner in which these periodic points depend on the parameter seS.Instead of discussing the general element y e F we define a subset ^S^F and discuss the periodic points for tpe'S.Then by showing 'S is a residual subset of F insures that the discussion dealt with the generic case.The problem discussed here is closely related to the following problems in the theory of differential equations.Consider Duffing's equation d2xjdt2 + x + x3

A continuous function on the product of compact metric spaces to itself and back to the same space is known as a genera system. Where each element's orbit is an infinite sequence and where the first two elements are the same as given and next to it depends on the two elements prior to it form stronger conditions for the orbit. We could define an m-step dynamical system by extending the definition of a compact metric space to its m-times product. Because the system's current state frequently depends directly on the conditions of previous terms, this system appears more realistic. The basic theorems regarding periodic points and their related points, such as fixed point, limit points, recurrent points, will be proved in this paper. We also define the topological transitivity and its properties. In the end we find periodic points with periods one and two for affine maps and periodicity of tent map in dynamical system and generalised dynamical system.

The realization problem that we will state here can be divided into the following three problems A, B and C. Where I/O is the set of input/output maps that may be the input/output relation of a given black-box. CD is the category of dynamical systems which may have the same behavior (equivalently, input/output relation) as the black-box. A. The existence and uniqueness in algebraic sense. For any input/output map a I/O , find out at least one dynamical system CD such that the behavior of it is a . Also prove that any two dynamical systems that have the same behavior a are isomorphic in the sense of the category CD. B. The finite dimensionality of the dynamical systems. Clarify when a dynamical system CD is finite dimensional. Because finite dimensional dynamical systems are actually appearing by linear (or non-linear) circuits or computer programs, it is very important that these conditions become clear. C. Deriving the dynamical systems from finite data. Partial realization problems are to find the minimal dynamical system fit to a given finite input/output’s data and to clarify when the minimal dynamical systems are isomorphic.

Complexity and proper quasi-weakly almost periodic points

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.

For a compact space K we denote by $$C_w(K)$$ ( $$C_p(K)$$ ) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that K is an infinite (metrizable) compact space. Can $$C_w(K)$$ and $$C_p(K)$$ be homeomorphic? We show that the answer is “no”, provided K is an infinite compact metrizable C-space. In particular our proof works for any infinite compact metrizable finite-dimensional space K.

We establish a sufficient condition for a continuous map, acting on a compact metric space, to have a Baire residual set of points exhibiting historic behavior (also known as irregular points). This criterion applies, for instance, to a minimal and non-uniquely ergodic map; to maps preserving two distinct probability measures with full support; to non-trivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose stable manifolds are dense, including Mañé and Shub examples of robustly transitive non-hyperbolic diffeomorphisms. This way, our unifying approach recovers a collection of known deep theorems on the genericity of the irregular set, for both additive and sub-additive potentials, and also provides a number of new applications.

We study here the invariance of topological distributionally scrambled sets for maps on uniform spaces (not necessarily compact or metrizable). For a uniformly continuous surjective self-map defined on a uniformly locally compact Hausdorff uniform space having topological weak specification, a fixed point, and countably many periodic points with distinct periods, we prove that the map admits an invariant topological distributionally scrambled set of type 1. Further, if the uniform space is second countable, the map admits a dense Mycielski invariant topological distributionally scrambled set of type 1.