Cell decomposition and definable functions for weak p‐adic structures

We develop a notion of cell decomposition suitable for studying weak p‐adic structures (reducts of p‐adic fields where addition and multiplication are not (everywhere) definable). As an example, we consider a structure with restricted addition.     

集值离散系统的某些动力性质

A discrete dynamical system can be expressed as xn 1 =f(xn), n=0,1, 2,... where X isa metric space and f : X→X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]).     

Unique Ergodicity for Zero-entropy Dynamical Systems with the Approximate Product Property

We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.     

拟Lipschitz等价的进展综述

讨论了拟Lipschitz等价的来源、进展与应用, 并提出了若干公开问题.     

Minimality and reductibility of conditionally poisson systems with finite state space

In this paper we consider stochastic systems with finite state space and counting process output. In particular we address the question whether a given system has a minimal representation, where roughly speaking minimality means minimality of the size of the state space. We show that minimality is connected to a suitably defined notion of observability. Finally we present an algorithm that enables us, starting from a given representation, to construct a minimal representation for the same system     

Sensitivity and regionally proximal relation in minimal systems

A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same moment.The properties of n-sensitivity in minimal systems are investigated.It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct.Moreover,the structure of a minimal system which is n-sensitive but not(n 1)-sensitive(n≥2)is determined.     

A note on stable sets,groups, and theories with NIP

Let M be an arbitrary structure. Then we say that an M ‐formula φ (x) defines a stable set in M if every formula φ (x) ∧ α (x, y) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G /H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure. More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Definable functions in the ℵ<Subscript>0</Subscript>-categorical weakly circularly minimal structures

We continue studying the analogs of o-minimality and weak o-minimality for circularly ordered sets. We present a complete characterization of the behavior of unary definable functions in an ?₀-categorical 1-transitive weakly circularly minimal structure. Using it, we describe the ?₀-categorical 1-transitive nonprimitive weakly circularly minimal structures of convexity rank greater than 1 up to binarity.     

A remark on Schrödinger operators on aperiodic tilings

We prove that for a large class of Schrödinger operators on aperiodic tilings the spectrum and the integrated density of states are the same for all tilings in the local isomorphism class, i.e., for all tilings in the orbit closure of one of the tilings. This generalizes the argument in earlier work from discrete strictly ergodic operators onl ²( ^d ) to operators on thel ²-spaces of sets of vertices of strictly ergodic tilings.