Minimal Dynamical Systems Research Articles

Mathematical Modeling as a Tool for Basic Research in Acupuncture

We propose and illustrate a five-step strategy for basic research of traditional explanatory frameworks of acupuncture. Our approach is based on using mathematical models as a bridge between the traditional explanatory frameworks of acupuncture and Western research technologies. Step 1. Carefully select and document case studies that would allow simultaneous interpretation within several "traditional" explanatory frameworks of acupuncture. Step 2. Develop minimal theoretical models connecting the diagnosis and treatment within the context of each of the selected explanatory frameworks of acupuncture. Step 3. Develop minimal dynamical systems models for each theoretical model from Step 2, so as to tighten their logical structure and to bring them into a falsifiable and more abstracted format (which provides the link between the theoretical models at Step 2 and possible Western-based models, and hence Western measurement technology). Step 4. Interpret the mathematical models from Step 3 within the framework of "Western" scientific perspectives. Step 5. Select and conduct appropriate "objective" (skin electrical impedance, etc.) measurements to test the validity of the models at Step 4. Repeat Steps 1-5 as needed.

Structural properties of universal minimal dynamical systems for discrete semigroups

We show that for a discrete semigroup S S there exists a uniquely determined complete Boolean algebra B ( S ) B(S) - the algebra of clopen subsets of M ( S ) M(S) . M ( S ) M(S) is the phase space of the universal minimal dynamical system for S S 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 B ( S ) B(S) is either atomic or atomless; that B ( S ) B(S) is weakly homogenous provided S S has a minimal left ideal; and that for countable semigroups B ( S ) B(S) 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 G ( S ) G(S) of a group-like semigroup S S can be constructed via universal minimal dynamical system for S S and, moreover, B ( S ) B(S) and B ( G ( S ) ) B(G(S)) are the same.

Decomposing isometric extensions using group extensions

AbstractThis paper studies the structure of isometric extensions of compact metric topological dynamical systems with ℤ action and gives two decompositions of the general case to a more structured case. Suppose that Y → X is a M-isometric extension. An extension, Z, of Y is constructed which is also a G-isometric extension of X, where G is the group of isometries of M. The first construction shows that, provided that (X, T) is transitive, there are almost-automorphic extensions Y′ → Y and X′ → X, so that Y′ is homeomorphic to X′ × M and the natural projection Y′ → X′ is a group extension. The second shows that, provided that (X, T) is minimal, there is a G-action on Z which commutes with T and which preserves fibres and acts on each of them minimally. Each individual orbit closure, Za, in Z is a G′-isometric extension of X, where G′ is a subgroup of G, and there is a G′-action on Za which commutes with T, preserves fibres and acts minimally on each of them. Two illustrations are presented. Of the first: to reprove a result of Furstenberg; that every distal point is IP*-recurrent. Of the second: to describe the minimal subsets in isometric extensions of minimal topological dynamical systems.

Algorithmic complexity of points in dynamical systems

AbstractThis work is based on the author's dissertation. We examine the algorithmic complexity (in the sense of Kolmogorov and Chaitin) of the orbits of points in dynamical systems. Extending a theorem of A. A. Brudno, we note that for any ergodic invariant probability measure on a compact dynamical system, almost every trajectory has a limiting complexity equal to the entropy of the system. We use these results to show that for minimal dynamical systems, and for systems with the tracking property (a weaker version of specification), the set of points whose trajectories have upper complexity equal to the topological entropy is residual. We give an example of a topologically transitive system with positive entropy for which an uncountable open set of points has upper complexity equal to zero. We use techniques from universal data compression to prove a recurrence theorem: if a compact dynamical system has a unique measure of maximal entropy, then any point whose lower complexity is equal to the topological entropy is generic for that unique measure. Finally, we discuss algorithmic versions of the theorem of Kamae on preservation of the class of normal sequences under selection by sequences of zero Kamae-entropy.

On equicontinuous factors of linear extensions of minimal dynamical systems

The concept of the equicontinuous factor of the linear extension of a minimal transformation group is introduced and investigated. It is shown that a subset of motions, bounded and distal with respect to the extension, forms a maximal equicontinuous subsplitting of the linear extension. As a consequence, any distal linear extension has a nontrivial equicontinuous invariant subsplitting. The linear extensions without exponential dichotomy possess similar subsplittings if the Favard condition is satisfied. The same statement holds for linear extensions with the property of recurrent motions additivity provided that at least one nonzero motion of this sort exists.

A state space isomorphism theorem for minimal dynamical systems

Consider a discrete time dynamical systemxk+1=f(x k ) on a compact metric spaceM, wheref:M→M is a continuous map. Leth:M→B k be a continuous output function. Suppose that all of the positive orbits off are dense and that the system is observable. We prove that any output trajectory of the system determinesf andh andM up to a homeomorphism. IfM is a compact Abelian topological group andf is an ergodic translation, then any output trajectory determines the system up to a translation and a group isomorphism of the group.

Observing Ergodic Translations on Compact Abelian Groups with Discontinuous Functions

Several authors have considered the problem of global observability for minimal dynamical systems, and particularly the example of ergodic (equivalently, minimal) translations on compact abelian groups. The global observability problem for ergodic translations is considered here in the case where the output function may be discontinuous. The result for continuous output functions, essentially due to McMahon, is described. This result shows that a continuous output function observes all ergodic translations if and only if it has no nontrivial symmetries. This result is extended here in two directions. First, functions continuous except on a meagre set and symmetries modulo meagre sets are considered. A result analogous to the continuous case is obtained that includes and generalizes the continuous case and results of Balogh, Bennett, and Martin on observing ergodic translations with the characteristic functions of certain subsets of the group. Functions continuous except on a set of measure zero are also considered. Secondly, ergodic theory and harmonic analysis techniques are employed to consider the case of intergrable output functions and symmetries modulo sets of measure zero. It is shown that an integrable function observes ergodic translations modulo sets of measure zero if and only if it has no nontrivial symmetries modulo sets of measure zero. The group of symmetries of the output function modulo sets of measure zero can be determined from its Fourier series. The results here show that classical observability can be determined from the Fourier series when the function is continuous almost everywhere, but only observability modulo sets of measure zero can be determined when the output function is merely integrable.

Skew products and minimal dynamical systems on separable Hilbert manifolds

AbstractWe prove that any locally compact, non-compact, second countable group acts minimally on any metrizable connected manifold modelled on the separable Hilbert space.

Toeplitz minimal flows which are not uniquely ergodic

We study minimal symbolic dynamical systems which are orbit closures of Toeplitz sequences. We construct 0–1 subshifts of this type for which the set of ergodic invariant measures has any given finite cardinality, is countably infinite or has cardinality of the continuum.

Simplicity of C* -Algebras of Minimal Dynamical Systems

Simplicity of C^* -Algebras of Minimal Dynamical Systems

The spectrum of dynamical systems arising from substitutions of constant length

Minimal flows and dynamical systems arising from substitutions are considered. In the case of substitutions of constant length the trace relation of the flow is calculated and is used to determine the spectrum of the dynamical system. Several methods are indicated to obtain new substitutions from given ones, leading among other things to a description of the behaviour of powers of the shift homeomorphism on the system arising from any substitution.

The Structure of Substitution Minimal Sets

Substitutions of constant length on two symbols and their corresponding minimal dynamical systems are divided into three classes: finite, discrete and continuous. Finite substitutions give rise to uninteresting systems. Discrete substitutions yield strictly ergodic systems with discrete spectra, whose topological structure is determined precisely. Continuous substitutions yield strictly ergodic systems with partly continuous and partly discrete spectra, whose topological structure is studied by means of an associated discrete substitution. Topological and measure-theoretic isomorphisms are studied for discrete and continuous substitutions, and a complete topological invariant, the normal form of a substitution, is given.

The structure of substitution minimal sets

Substitutions of constant length on two symbols and their corresponding minimal dynamical systems are divided into three classes: finite, discrete and continuous. Finite substitutions give rise to uninteresting systems. Discrete substitutions yield strictly ergodic systems with discrete spectra, whose topological structure is determined precisely. Continuous substitutions yield strictly ergodic systems with partly continuous and partly discrete spectra, whose topological structure is studied by means of an associated discrete substitution. Topological and measure-theoretic isomorphisms are studied for discrete and continuous substitutions, and a complete topological invariant, the normal form of a substitution, is given. 0. Introduction. Let 0 denote a substitution of constant length n on the symbols 0 and 1. Our main objective is the classification of the dynamical systems (0,, r) arising from such substitutions. A substitution 0 is either finite, discrete, or continuous as defined in ?3. Finite substitutions give rise to dynamical systems all of whose minimal sets are periodic orbits. Discrete and continuous substitutions both give rise to strictly ergodic dynamical systems with discrete and partially continuous spectra respectively. More specifically, if 0 is discrete, then V. is the orbit-closure of a Toeplitz bisequence, and if 0 is continuous, then (C is the orbit-closure of an extended generalized Morse sequence. Dynamical systems belonging to discrete substitutions are measure-theoretically isomorphic if and only if the lengths of the substitutions have the same prime factors. If 0 is discrete, we are able to give an explicit construction of (0,, r) as an almost one-to-one extension of the n-adic system (Z(n), r). We associate with 0 an object called a group system which is a complete invariant for topological isomorphism of (0,, r). The concept of normal form for a discrete substitution is defined and it is shown that discrete substitutions of the same length possess topologically isomorphic dynamical systems if and only if they have the same normal form. To each continuous substitution 0 we associate a discrete substitution 0 of the same length such that (0, r) is a distal extension of (06, r). Each fibre consists of two points which are mirror images of each other. A normal form for continuous Received by the editors May 22, 1970. AMS 1970 subject classifications. Primary 28A65, 54H20.

Minimal Dynamical Systems with Quasi-Discrete Spectrum

Minimal Dynamical Systems with Quasi-Discrete Spectrum

On the Iteration of Transformations in Noncompact Minimal Dynamical Systems

On the Iteration of Transformations in Noncompact Minimal Dynamical Systems

On the iteration of transformations in noncompact minimal dynamical systems

Let A be a Hausdorff space, 4 a continuous mapping of A into itself. It is the purpose of the present paper to discuss various topics centering around the following question: If g is a bounded continuous function on A, does there exist a bounded continuous function f on A such that f(ba) -f(a) =g(a) for all a in A? Suppose that for each ao in A, the set {Ionao, nO} is dense in A. Theorem 1 asserts that a necessary and sufficient condition for the existence of such an f is that Z k0 g(40ka) I should be uniformly bounded for all positive j and all points a of A. For homeomorphisms of compact spaces, this result was previously obtained by Gottschalk and Hedlund [5, Theorem 14.11, p. 135].1 A related problem for linear operators in a Banach space is obtained by letting X be the Banach space of bounded continuous functions on A with the uniform norm, T the linear transformation of X into itself defined by (Tf)(a) =f(Oa), aEEA. In terms of X and T, Theorem 1 states that g will lie in the range of (IT) if and only if the sequence of norms || Dk=o Tkglj is uniformly bounded for all positive j. In a reflexive Banach space, this characterization of the range of (IT) is valid for any linear transformation T for which || Tnj1 is bounded for all n. A sufficient condition in a general Banach space would seem to require an assumption that the elements EJ=o Tkg } lie for all j in a fixed weakly compact subset K of X. It would be interesting to obtain a proof of Theorem 1 along these lines. We shall content ourselves with showing by these mnethods that if m is a totally-finite measure on a a--algebra on A, Lo(m) the space of m-essentially bounded measurable functions, 4 a measure preserving mapping of A into A, then in order that for an element g in Lo(m), there should exist anf in Lc(m) such thatf(ka) -f(a) =g(a) a.e. in m, it is necessary and sufficient that m-ess. sup. | =0 g(Qka) | should be uniformly bounded for all positive j.2 Such a result raises another sort of question. For a topological space A if g is continuous and f is a solution of the equation f(ba)

On spaces which are not of countable character

O.N.R. Research Contract Nonr-222(37), Dept. of Mathematics, University of California at Berkeley, 1957. 4. N. Dunford and J. T. Schwartz, Linear operators, vol. 1, Chapter 5, to appear. 5. W. H. Gottschalk and G. A. Hedlund, Topological dynamics, Amer. Math. Soc. Colloquium Publications, New York, 1955. 6. G. A. Hedlund, A class of transformations of the plane, Proc. Cambridge Philos. Soc. vol. 51 (1955) pp. 554-564. 7. S. Kakutani, On the continuity of eigenfunctions of minimal dynamical systems, to appear. 8. K. Yosida and S. Kakutani, Operator theoretical treatment of Markoff's process and mean ergodic theorem, Ann. of Math. vol. 42 (1941) pp. 188-228.