4,823 research outputs found
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 is a \s-finite -invariant measure
having the property that one can find a real number so that
but ,
where is the first passage time function in the reference state 1. In
particular we shall consider invariant measures arising from a potential
which is uniformly continuous but not of summable variation. If then
can be normalized to give the unique non-atomic equilibrium probability
measure of for which we compute the (asymptotically) exact mixing rate, of
order . We also establish the weak-Bernoulli property and a polynomial
cluster property (decay of correlations) for observables of polynomial
variation. If instead then is an infinite measure with scaling
rate of order . 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 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 of
ergodic degree the rate of convergence towards the stationary
distribution is subgeometric of order , 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 on the Banach space . 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
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
On the generic triangle group
We introduce the concept of a generic Euclidean triangle and study the
group generated by the reflection across the edges of . In
particular, we prove that the subgroup of all translations in
is free abelian of infinite rank, while the index 2 subgroup of all
orientation preserving transformations in is free metabelian of rank
2, with as the commutator subgroup. As a consequence, the group
cannot be finitely presented and we provide explicit minimal infinite
presentations of both and . 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
holding for given non-generic triangles .Comment: 21 pages, 6 figure
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
An asymptotic dimension for metric spaces, and the 0-th Novikov-Shubin invariant
A nonnegative number d_infinity, called asymptotic dimension, is associated
with any metric space. Such number detects the asymptotic properties of the
space (being zero on bounded metric spaces), fulfills the properties of a
dimension, and is invariant under rough isometries. It is then shown that for a
class of open manifolds with bounded geometry the asymptotic dimension
coincides with the 0-th Novikov-Shubin number alpha_0 defined previously
(math.OA/9802015, cf. also math.DG/0110294). Thus the dimensional
interpretation of alpha_0 given in the mentioned paper in the framework of
noncommutative geometry is established on metrics grounds. Since the asymptotic
dimension of a covering manifold coincides with the polynomial growth of its
covering group, the stated equality generalises to open manifolds a result by
Varopoulos.Comment: 17 pages, to appear on the Pacific Journal of Mathematics. This paper
roughly corresponds to the third section of the unpublished math.DG/980904
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