Topological conjugacy to given constant length substitution minimal systems

We find necessary and sufficient conditions for a symbolic dynamical system to be topologically conjugate to any given constant length substitution minimal system, thus extending the results in Coven et al. (2008) for the Morse and Toeplitz substitutions.     

Morse decompositions of nonautonomous dynamical systems

The global asymptotic behavior of dynamical systems on compact metric spaces can be described via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections of attractors and repellers of the system. In this paper, new notions of attractor and repeller for nonautonomous dynamical systems are introduced which are designed to establish nonautonomous generalizations of the Morse decomposition. The dynamical properties of these decompositions are discussed, and nonautonomous Lyapunov functions which are constant on the Morse sets are constructed explicitly. Moreover, Morse decompositions of one-dimensional and linear systems are studied.

     

All-time Morse decompositions of linear nonautonomous dynamical systems

Morse decompositions provide inside information about the global asymptotic behavior of dynamical systems on compact metric spaces. Recently, the existence of Morse decompositions for nonautonomous dynamical systems was proved by restricting attention to the past or the future of the system, but in general, such a construction is not realizable for the entire time. In this article, it is shown that all-time Morse decompositions can be defined for linear systems on the projective space. Moreover, the dynamical properties are discussed and an analogue to the Theorem of Selgrade is proved.

     

赋予hit—or-miss拓扑超空间动力系统的拓扑熵

设（X,d,f）为拓扑动力系统,其中X为局部紧第二可数Hausdorff空间,d为紧型度量,f为完备映射,用2^x和f分别表示由X的所有非空闭子集和所有闭子集构成的集族,（2^x,ρ,2^f）和（f,ρ,2^f）为由（X,d,f）诱导的赋予hit—or—miss拓扑的超空间动力系统．本文研究了h（X,d,f）和h（2^...     

Topological dynamic consistency of non-standard finite difference schemes for dynamical systems

This work expands the mathematical theory which connects continuous dynamical systems and the discrete dynamical systems obtained from the associated numerical schemes. The problem is considered within the setting of Topological Dynamics. The topological dynamic consistency of a family of DDSs and the associated continuous system is defined as topological equivalence between the evolution operator of the continuous system and the set of maps defining the respective DDSs, for all positive time-step sizes. The one-dimensional theory is developed and a few important representative examples are studied in detail. It is found that the design of non-standard topologically dynamically consistent schemes requires some care.     

Inverse generating function approach for the preconditioning of Toeplitz‐block systems

As proposed by R. H. Chan and M. K. Ng (1993), linear systems of the form T [ f ] x = b , where T [ f ] denotes the n×n Toeplitz matrix generated by the function f, can be solved using iterative solvers with as a preconditioner. This article aims at generalizing this approach to the case of Toeplitz‐block matrices and matrix‐valued generating functions F . We prove that if F is Hermitian positive definite, most eigenvalues of the preconditioned matrix T [ F ⁻¹]T[ F ] are clustered around one. Numerical experiments demonstrate the performance of this preconditioner.     

关于非自治动力系统中的h-极小覆盖

设（X,d1,f1∞）与（Y ,d2,g1,∞）为两个非自治动力系统,h是从（X,d1,f．∞）到（Y,d2,g1∞）的拓扑半共轭．通过对自治动力系统中的h一极小覆盖的研究,本文得到了以下结论：1）对于任意的Y∈Y及X∈h-1（y）,orb（x,f1∞）被h映射为orb（y,g1∞）,w（x,f1∞）被h映射为w（y,g1∞）;2）在（X,d1,f1∞）中引入关于拓扑半共轭的h-极小覆盖的定义,证明了h一极小覆盖的存在性;3）对于任意的XEX和Y∈Y,在（w（z,f1∞）,f1∞。（x,f1,∞）与（w（y,g∞）,g1,∞（y,g1∞））均构成原系统的子系统的前提下,R（f1∞）被h映射为R（g1∞）．这些结论丰富了非自治动力系统的内容．     

Combinatorics of sequential dynamical systems

In this paper we study sequential dynamical systems (SDS) over words. Our main result is the classification of SDS over words for fixed graph Y and family of local maps (F_vi) by means of a novel notion of SDS equivalence. This equivalence arises from a natural group action on acyclic orientations. An SDS consists of: (a) a graph Y, (b) a family of vertex indexed Y-local maps F_vi:Kⁿ→Kⁿ, where K is a finite field and (c) a word w, i.e. a family (w₁,…,w_k), where w_j is a Y-vertex. A map F_vi(x_v1,…,x_vn) is called Y-local iff it fixes all variables x_vj≠x_vi and depends exclusively on the variables x_vj, for v_j∈B₁(v_i). The SDS-map is obtained by composing the local maps F_vi according to the word w: . Mutual dependencies of the local maps arising from their sequential application are expressed in the graph G(w,Y) having vertex set {1,…,k} (the indices of the word w) and in which r,s are adjacent iff w_s,w_r are adjacent in Y. We prove a bijection from equivalence classes of SDS-words into equivalence classes of acyclic orientations of G(w,Y). We show that within these equivalence classes the induced SDS are equivalent in the sense that their respective phase spaces are isomorphic as digraphs.     

On the modulus of continuity for spectral measures in substitution dynamics

The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the “fractal” structure of these measures. The main results are, first, a Hölder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hölder estimate for self-similar suspension flows; and, third, a Hölder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hölder estimates for the spectral measures of translation flows along stable foliations of pseudo-Anosov automorphisms. A key technical tool in the proof of the second result is an “arithmetic-Diophantine” proposition, which has other applications. In Appendix A this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions.     

Structural properties of universal minimal dynamical systems for discrete semigroups

We show that for a discrete semigroup there exists a uniquely determined complete Boolean algebra - the algebra of clopen subsets of . is the phase space of the universal minimal dynamical system for and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that is either atomic or atomless; that is weakly homogenous provided has a minimal left ideal; and that for countable semigroups is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reflection of a group-like semigroup can be constructed via universal minimal dynamical system for and, moreover, and are the same.

     

Rest points of generalized dynamical systems

In this paper we consider generalized dynamical systems whose integral vortex (that is, the set of all trajectories of the system starting at a given point) is an acyclic set in the corresponding space of curves. For such systems we apply the theory of fixed points for multi-valued maps in order to prove the existence of rest points. In this way we obtain new existence theorems for rest points of generalized dynamical systems. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 28–36, January, 1999.     

Two notes on measure-theoretic entropy of random dynamical systems

In this paper, Brin-Katok local entropy formula and Katok＇s definition of the measuretheoretic entropy using spanning set are established for the random dynamical system over an invertible ergodic system.     

Local singularities of dynamical systems with shock interactions

Local (qualitative) singularities of a special class of dynamical systems with shock interactions are classified. For the first four of the specified types of local singularities, certain properties of the qualitative structure are described and used to establish the topological equivalence of the corresponding singularities.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 531–542, October, 1998.     

Update schedules of sequential dynamical systems

Sequential dynamical systems have the property, that the updates of states of individual cells occur sequentially, so that the global update of the system depends on the order of the individual updates. This order is given by an order on the set of vertices of the dependency graph. It turns out that only a partial suborder is necessary to describe the global update. This paper defines and studies this partial order and its influence on the global update function.     

Reverse-engineering of polynomial dynamical systems

Multivariate polynomial dynamical systems over finite fields have been studied in several contexts, including engineering and mathematical biology. An important problem is to construct models of such systems from a partial specification of dynamic properties, e.g., from a collection of state transition measurements. Here, we consider static models, which are directed graphs that represent the causal relationships between system variables, so-called wiring diagrams. This paper contains an algorithm which computes all possible minimal wiring diagrams for a given set of state transition measurements. The paper also contains several statistical measures for model selection. The algorithm uses primary decomposition of monomial ideals as the principal tool. An application to the reverse-engineering of a gene regulatory network is included. The algorithm and the statistical measures are implemented in Macaulay 2, and are available from the authors.     

Topological Entropy of a Class of Subshifts of Finite Type

In this paper, we construct a special class of subshifts of finite type. By studying the spectral radius of the transfer matrix associated with the subshift of finite type, we obtain an estimation of its topological entropy. Interestingly, we find that the topological entropy of this class of subshifts of finite type converges monotonically to log(n + 1) (a constant only depends on the structure of the transfer matrices) as the increasing of the order of the transfer matrices.     

Asymptotics of solutions of infinite-dimensional homogeneous dynamical systems

In this paper we study the connection between the uniform asymptotic stability and the power-law or exponential asymptotics of the solutions of infinite-dimensional systems (differential equations in Banach spaces, functional differential equations, and completely solvable multidimensional differential equations). Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 115–126, January, 1998.