Sensitivity for topologically double ergodic dynamical systems

<abstract><p>As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.</p></abstract>

Fuzzy Fibrewise Homotopy

In this disquisition, the notion structures is introduced. The intend of this article is to study the notions of homotopy, path homotopy and fundamental group via structure which is a triplet consisting of two fuzzy topological spaces and a continuous surjection between them. Many properties concerning these concepts are provided.

Equivalent conditions for digital covering maps

It is known that every digital covering map p:(E,k) ? (B,?) has the unique path lifting property. In this paper, we show that its inverse is true when the continuous surjective map p has no conciliator point. Also, we prove that a digital (k,?)-continuous surjection p:(E,k)? (B,?) is a digital covering map if and only if it is a local isomorphism, when all digital spaces are connected. Moreover, we find out a loop criterion for a digital covering map to be a radius n covering map.

The function ω ƒ on simple n-ods

<p>Given a discrete dynamical system (X, ƒ), we consider the function ω_ƒ-limit set from X to 2^xas</p><p>ω_ƒ(x) = {y ∈ X : there exists a sequence of positive integers <br /> n₁ &lt; n₂ &lt; … such that lim_k_→_∞ ƒ^nk (x) = y},</p><p>for each x ∈ X. In the article [1], A. M. Bruckner and J. Ceder established several conditions which are equivalent to the continuity of the function ω_ƒ where ƒ: [0,1] → [0,1] is continuous surjection. It is natural to ask whether or not some results of [1] can be extended to finite graphs. In this direction, we study the function ω_ƒ when the phase space is a n-od simple T. We prove that if ω_ƒ is a continuous map, then Fix(ƒ²) and Fix(ƒ³) are connected sets. We will provide examples to show that the inverse implication fails when the phase space is a simple triod. However, we will prove that:</p><p>Theorem A 2. If ƒ: T → T is a continuous function where T is a simple triod then ω_ƒ is a continuous set valued function iff the family {ƒ⁰, ƒ¹, ƒ²,} is equicontinuous.</p><p>As a consequence of our results concerning the ω_ƒ function on the simple triod, we obtain the following characterization of the unit interval.</p><p>Theorem A 1. Let G be a finite graph. Then G is an arc iff for each continuous function ƒ: G → G the following conditions are equivalent:<br /> (1) The function ω_ƒ is continuous.<br /> (2) The set of all fixed points of ƒ²is nonempty and connected.</p>

ABSTRACT ω-LIMIT SETS

The shift map σ on ω* is the continuous self-map of ω* induced by the function n ↦ n + 1 on ω. Given a compact Hausdorff space X and a continuous function f : X → X, we say that (X, f) is a quotient of (ω*, σ) whenever there is a continuous surjection Q : ω*→ X such that Q ○ σ = σ ○ f.Our main theorem states that if the weight of X is at most ℵ1, then (X, f) is a quotient of (ω*, σ), if and only if f is weakly incompressible (which means that no nontrivial open U ⊆ X has $f\left( {\bar{U}} \right) \subseteq U$). Under CH, this gives a complete characterization of the quotients of (ω*, σ) and implies, for example, that (ω*, σ−1) is a quotient of (ω*, σ).In the language of topological dynamics, our theorem states that a dynamical system of weight ℵ1 is an abstract ω-limit set if and only if it is weakly incompressible.We complement these results by proving (1) our main theorem remains true when ℵ1 is replaced by any κ < p, (2) consistently, the theorem becomes false if we replace ℵ1 by ℵ2, and (3) OCA + MA implies that (ω*, σ−1) is not a quotient of (ω*, σ).

From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra

We use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered groupGwith a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriateG-indexed zero-dimensional compactification{w_{G}Z_{G}}of its space{Z_{G}}ofminimalprime ideals. The two further ingredients needed to establish this representation are the Yosida representation ofGon its space{X_{G}}ofmaximalideals, and the well-known continuous surjection of{Z_{G}}onto{X_{G}}. We then establish our main result by showing that the inclusion-minimal extension of this representation ofGthat separates the points of{Z_{G}}– namely, the sublattice subgroup of{\operatorname{C}(Z_{G})}generated by the image ofGalong with all characteristic functions of clopen (closed and open) subsets of{Z_{G}}which are determined by elements ofG– is precisely the classical projectable hull ofG. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.

Relating Different Cycle Spaces of the Same Infinite Graph

Casteels and Richter have shown that if $X$ and $Y$ are distinct compactifications of a locally finite graph $G$ and $f:X\to Y$ is a continuous surjection such that $f$ restricts to a homeomorphism on $G$, then the cycle space $Z_X$ of $X$ is contained in the cycle space $Z_Y$ of $Y$. In this work, we show how to extend a basis for $Z_X$ to a basis of $Z_Y$.

Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the ‘overlap set’ 𝒪 is finite, and which are ‘invertible’ on the attractor A, in the sense that there is a continuous surjection q:A→A whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that q is not a local homeomorphism precisely at 𝒪. We suppose also that there is a rational function p with the Julia set J such that (A,q) and (J,p) are conjugate. We prove that if A has bounded turning and p has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS {λz,λz+1} where λ is a complex parameter in the unit disk, such that its attractor Aλ is a dendrite, which happens whenever 𝒪 is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map qλ on Aλ. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map pc (z)=z2 +c, with the Julia set Jc such that (Aλ,qλ) and (Jc,pc) are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.

NONCOMMUTATIVE IMAGES OF COMMUTATIVE SPECTRA

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially "sewn together" to form Spec R. In particular, we construct a bimodule-determined functor Mod Z → Mod R, for a suitable commutative noetherian ring Z, from which there follows a finite-to-one, continuous surjection Spec Z → Spec R. Algebras satisfying the given axiomatic framework include PI algebras finitely generated over fields, noetherian PI algebras, enveloping algebras of complex finite dimensional solvable Lie algebras, standard generic quantum semisimple Lie groups, quantum affine spaces, quantized Weyl algebras, and standard generic quantizations of the coordinate ring of n × n matrices. In all of these examples (except for the non-finitely-generated noetherian PI algebras), Z is finitely generated over a field, and the constructed map of spectra restricts to a surjection Max Z → Prim R.