For a T1 topological space X, Zhao and Xi constructed a dcpo model Pˆ, where P is a bounded complete algebraic poset model of X. This paper explores how some topological properties related to sobriety are preserved under Xi-Zhao model. We formulate the closed WD subsets of the maximal point space Max(Pˆ) and the Scott space ΣPˆ, and then prove that X is a WD space if and only if ΣPˆ is a WD space. We also show that the sobrification Xs (resp., the well-filtered reflection Xw) of X can be embedded into the sobrification Pˆs (resp., the well-filtered reflection Pˆw) of Pˆ as a saturated subspace. Finally, we introduce two new concepts “H-model spaces” and “weak H-model spaces”, and provide a general framework to prove that such topological properties are also preserved under Xi-Zhao models.
In this paper, we answer some questions about dynamical properties similar to transitivity, sometimes called variations of transitivity. Three of these questions appear in the articles [7] and [3]. The most important results in this paper are the following. First, let n∈N, and for each 1≤i≤n, let (Xi,fi) be a dynamical system. We show that if the dynamical system induced in the symmetric product (Fn(∏i=1nXi),Fn(∏i=1nfi)) is strongly product transitive, then for each 1≤i≤n, the dynamical system (Fn(Xi),Fn(fi)) is strongly product transitive. Second, we prove that if (Fn(∏i=1nXi),Fn(∏i=1nfi)) is fully exact and each fi is continuous, then, for each 1≤i≤n, the dynamical system (Fn(Xi),Fn(fi)) is also fully exact. Finally, we present an example of a dynamical system (X,f), with f a continuous function, such that f×2 is strongly transitive, but f×3 is not.
We introduce weighted subsequential distributional chaos (WS–DC) for dynamical systems on symbolic spaces. A gauge consists of a weighted coordinate specification and a sequence of observation times, while the associated coordinatewise pseudometric serves only as a continuous observable on the original compact metric system.We prove that every DC1 system on a symbolic space is WS–DC and that, for subshifts under full-time sampling, a pair is scrambled for some gauge if and only if it is DC1-scrambled; in particular, full-time WS–DC is equivalent to DC1. Thus, any strict enlargement of DC1 arises from subsequential sampling rather than coordinate weighting. By allowing arbitrary continuous pseudometrics, we obtain a generalized notion invariant under topological conjugacy.We apply this framework to carrying attractors, which are conjugate to n-adic adding machines. Such attractors admit no generalized distributionally scrambled pair under full-time sampling. This obstruction is sharp: although neither a metric nor a convergent sampling scheme can scramble a pair in an adding machine, a degenerate coordinate pseudometric combined with a suitable sparse time sequence scrambles an uncountable set. When n is not a power of 2, the ambient interval map nevertheless has positive topological entropy and admits an uncountable DC1-scrambled set disjoint from the attractor. We also establish the renormalization identity and the associated tower of invariant return domains for the generalized Feigenbaum functional equation.
Using an idea of Preiss–Vilímovský in [11], insertion theorems for (uniformly) continuous monotone functions are proved. Extension theorems for monotone functions then follow.
A self-map T of a metric space is graphic nonexpansive ifd(Tx,T2x)≤d(x,Tx)(x∈X). We call a metric space graphically plastic if every bijective graphic nonexpansive self-map preserves the distances between consecutive iterates. We characterize these spaces by injective bilateral sequences with nonincreasing consecutive distances. It follows that graphic plasticity passes to subspaces and that every graphically plastic metric space is totally disconnected. We prove that a compact ultrametric space is graphically plastic if and only if it has at most one accumulation point. For continuous graphic nonexpansive maps, equality holds at every recurrent point. We compare graphic plasticity with ordinary plasticity and show that neither property implies the other.
