On systems with finite ergodic degree

In this paper we study the ergodic theory of a class of symbolic dynamical systems (\O, T, \mu) where T:{\O}\to \O the left shift transformation on \O=\prod_0^\infty\{0,1\} and $\mu$ is a \s-finite $T$-invariant measure having the property that one can find a real number $d$ so that $\mu(\tau^d)=\infty$ but $\mu(\tau^{d-\epsilon})0$, where $\tau$ is the first passage time function in the reference state 1. In particular we shall consider invariant measures $\mu$ arising from a potential $V$ which is uniformly continuous but not of summable variation. If $d>0$ then $\mu$ can be normalized to give the unique non-atomic equilibrium probability measure of $V$ for which we compute the (asymptotically) exact mixing rate, of order $n^{-d}$. We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead $d\leq 0$ then $\mu$ is an infinite measure with scaling rate of order $n^d$. Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operator-valued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their non-polar singularity at $z=1$ is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point.Comment: 42 page

Infinite-volume mixing for dynamical systems preserving an infinite measure

In the scope of the statistical description of dynamical systems, one of the defining features of chaos is the tendency of a system to lose memory of its initial conditions (more precisely, of the distribution of its initial conditions). For a dynamical system preserving a probability measure, this property is named `mixing' and is equivalent to the decay of correlations for observables in phase space. For the class of dynamical systems preserving infinite measures, this probabilistic connection is lost and no completely satisfactory definition has yet been found which expresses the idea of losing track of the initial state of a system due to its chaotic dynamics. This is actually on open problem in the field of infinite ergodic theory. Virtually all the definitions that have been attempted so far use "local observables", that is, functions that essentially only "see" finite portions of the phase space. In this note we introduce the concept of "global observable", a function that gauges a certain quantity throughout the phase space. This concept is based on the notion of infinite-volume average, which plays the role of the expected value of a global observable. Endowed with these notions, whose rigorous definition is to be specified on a case-by-case basis, we give a number of definitions of infinite mixing. These fall in two categories: global-global mixing, which expresses the "decorrelation" of two global observables, and global-local mixing, where a global and a local observable are considered instead. These definitions are tested on two types of infinite-measure-preserving dynamical systems, the random walks and the Farey map.Comment: 15 pages, 3 figure

On the rate of convergence to equilibrium for countable ergodic Markov chains

Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between convergence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Markov operator $P$ on the Banach space $\ell_1$. Explicit conditions allowing to obtain the actual asymptotics for the rate of convergence are also discussed.Comment: 31 pages. to appear in Markov Processes and Related Field

On a set of numbers arising in the dynamics of unimodal maps

In this paper we initiate the study of the arithmetical properties of a set numbers which encode the dynamics of unimodal maps in a universal way along with that of the corresponding topological zeta function. Here we are concerned in particular with the Feigenbaum bifurcation.Comment: 12 page

REASONING DISPARITIES BETWEEN HK AND US MANAGERS

The purpose of this study was to examine and compare the factors that influence intuition as a decision-making tool for leaders/managers in Hong Kong and in the United States. This study examined the relationships among gender, management level, extent of management experience, country of operation, and the reported use of intuition in decision making. Existing empirical research in this field is sparse. In this research, attempt was made to contribute to empirical research on the viability and reported use of intuition as a decision-making skill of leaders. Agor’s Intuitive Measurement Survey (AIM) survey was adapted (with permission from copyright owner) from Weston Agor’s study to measure the relationship between a manager’s reported use of intuition in decision making and the manager’s management level, his level of management experience, the manager’s gender, and the manager’s country of operation. Each participant was electronically sent a link that led to a web page containing the survey questions. Once the respondent clicks submit, the questionnaire was mailed directly to the researcher. The research shows significant relationship between research variables. Administrative managers in Hong Kong’s reported use of intuition in decision making was significantly lower than US managers reported use of intuition in decision making. The paper concludes by examining the implications of these significant findings to global business management and management education.comparative management, management styles

On the generic triangle group

We introduce the concept of a generic Euclidean triangle $\tau$ and study the group $G_\tau$ generated by the reflection across the edges of $\tau$. In particular, we prove that the subgroup $T_\tau$ of all translations in $G_\tau$ is free abelian of infinite rank, while the index 2 subgroup $H_\tau$ of all orientation preserving transformations in $G_\tau$ is free metabelian of rank 2, with $T_\tau$ as the commutator subgroup. As a consequence, the group $G_\tau$ cannot be finitely presented and we provide explicit minimal infinite presentations of both $H_\tau$ and $G_\tau$. This answers in the affirmative the problem of the existence of a minimal presentation for the free metabelian group of rank 2. Moreover, we discuss some examples of non-trivial relations in $T_\tau$ holding for given non-generic triangles $\tau$.Comment: 21 pages, 6 figure

Noncommutative Riemann integration and and Novikov-Shubin invariants for open manifolds

Given a C*-algebra A with a semicontinuous semifinite trace tau acting on the Hilbert space H, we define the family R of bounded Riemann measurable elements w.r.t. tau as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that R is a C*-algebra, and tau extends to a semicontinuous semifinite trace on R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A'' and can be approximated in measure by operators in R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a tau-a.e. bimodule on R, denoted by R^, and such bimodule contains the functional calculi of selfadjoint elements of R under unbounded Riemann measurable functions. Besides, tau extends to a bimodule trace on R^. Type II_1 singular traces for C*-algebras can be defined on the bimodule of unbounded Riemann-measurable operators. Noncommutative Riemann integration, and singular traces for C*-algebras, are then used to define Novikov-Shubin numbers for amenable open manifolds, show their invariance under quasi-isometries, and prove that they are (noncommutative) asymptotic dimensions.Comment: 34 pages, LaTeX, a new section has been added, concerning an application to Novikov-Shubin invariants, the title changed accordingl

Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple (A,D,H), the functionals on A of the form a -> tau_omega(a|D|^(-t)) are studied, where tau_omega is a singular trace, and omega is a generalised limit. When tau_omega is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|^(-1), and that the set of t's for which there exists a singular trace tau_omega giving rise to a non-trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The corresponding functionals are called Hausdorff-Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit omega.Comment: latex, 36 pages, no figures, to appear on Journ. Funct. Analysi

The problem of completeness for Gromov-Hausdorff metrics on C*-algebras

It is proved that the family of equivalence classes of Lip-normed C*-algebras introduced by M. Rieffel, up to isomorphisms preserving the Lip-seminorm, is not complete w.r.t. the matricial quantum Gromov-Hausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely order isomorphic to any C*-algebra. Conditions ensuring the existence of a C*-structure on the limit are considered, making use of the notion of ultraproduct. More precisely, a necessary and sufficient condition is given for the existence, on the limiting operator system, of a C*-product structure inherited from the approximating C*-algebra. Such condition can be considered as a generalisation of the f-Leibniz conditions introduced by Kerr and Li. Furthermore, it is shown that our condition is not necessary for the existence of a C*-structure tout court, namely there are cases in which the limit is a C*-algebra, but the C*-structure is not inherited.Comment: 31 pages. Accepted for publication in Journal of Functional Analysi