2,767 research outputs found
Bowen Measure From Heteroclinic Points
We present a new construction of the entropy-maximizing, invariant
probability measure on a Smale space (the Bowen measure). Our construction is
based on points that are unstably equivalent to one given point, and stably
equivalent to another: heteroclinic points. The spirit of the construction is
similar to Bowen's construction from periodic points, though the techniques are
very different. We also prove results about the growth rate of certain sets of
heteroclinic points, and about the stable and unstable components of the Bowen
measure. The approach we take is to prove results through direct computation
for the case of a Shift of Finite type, and then use resolving factor maps to
extend the results to more general Smale spaces
On embedding of the Bratteli diagram into a surface
We study C*-algebras O_{\lambda} which arise in dynamics of the interval
exchange transformations and measured foliations on compact surfaces. Using
Koebe-Morse coding of geodesic lines, we establish a bijection between Bratteli
diagrams of such algebras and measured foliations. This approach allows us to
apply K-theory of operator algebras to prove strict ergodicity criterion and
Keane's conjecture for the interval exchange transformations.Comment: final versio
Families of type III KMS states on a class of C-algebras containing On and QN
We construct a family of purely infinite C¤-algebras, Q¸ for ¸ 2 (0, 1) that are classified by their K-groups. There is an action of the circle T with a unique KMS state à on each Q¸. For ¸ = 1/n, Q1/n »= On, with its usual T action and KMS state. For ¸ = p/q, rational in lowest terms, Q¸ »= On (n = q − p + 1) with UHF fixed point algebra of type (pq)1. For any n \u3e 1, Q¸ »= On for infinitely many ¸ with distinct KMS states and UHF fixed-point algebras. For any ¸ 2 (0, 1), Q¸ 6= O1. For ¸ irrational the fixed point algebras, are NOT AF and the Q¸ are usually NOT Cuntz algebras. For ¸ transcendental, K1(Q¸) »= K0(Q¸) »= Z1, so that Q¸ is Cuntz\u27 QN, [Cu1]. If ¸ and ¸−1 are both algebraic integers, the only On which appear are those for which n ´ 3(mod 4). For each ¸, the representation of Q¸ defined by the KMS state à generates a type III¸ factor. These algebras fit into the framework of modular index theory / twisted cyclic theory of [CPR2, CRT] and [CNNR]
The AF structure of non commutative toroidal Z/4Z orbifolds
For any irrational theta and rational number p/q such that q|qtheta-p|<1, a
projection e of trace q|qtheta-p| is constructed in the the irrational rotation
algebra A_theta that is invariant under the Fourier transform. (The latter is
the order four automorphism U mapped to V, V mapped to U^{-1}, where U, V are
the canonical unitaries generating A_theta.) Further, the projection e is
approximately central, the cut down algebra eA_theta e contains a Fourier
invariant q x q matrix algebra whose unit is e, and the cut downs eUe, eVe are
approximately inside the matrix algebra. (In particular, there are Fourier
invariant projections of trace k|qtheta-p| for k=1,...,q.) It is also shown
that for all theta the crossed product A_theta rtimes Z_4 satisfies the
Universal Coefficient Theorem. (Z_4 := Z/4Z.) As a consequence, using the
Classification Theorem of G. Elliott and G. Gong for AH-algebras, a theorem of
M. Rieffel, and by recent results of H. Lin, we show that A_theta rtimes Z_4 is
an AF-algebra for all irrational theta in a dense G_delta.Comment: 35 page
Mind and body, form and content: how not to do petitio principii analysis
Few theoretical insights have emerged from the extensive literature discussions of petitio principii argument. In particular, the pattern of petitio analysis has largely been one of movement between the two sides of a dichotomy, that of form and content. In this paper, I trace the basis of this dichotomy to a dualist conception of mind and world. I argue for the rejection of the form/content dichotomy on the ground that its dualist presuppositions generate a reductionist analysis of certain concepts which are central to the analysis of petitio argument. I contend, for example, that no syntactic relation can assimilate within its analysis the essentially holistic nature of a notion like justification. In this regard, I expound a form of dialectical criticism which has been frequently employed in the philosophical arguments of Hilary Putnam. Here the focus of analysis is upon the way in which the proponent of a position proceeds to explain or argue for his/her own particular theses. My conclusion points to the use of such dialectic within future analyses of petitio principii
Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice
`Umbral calculus' deals with representations of the canonical commutation
relations. We present a short exposition of it and discuss how this calculus
can be used to discretize continuum models and to construct representations of
Lie algebras on a lattice. Related ideas appeared in recent publications and we
show that the examples treated there are special cases of umbral calculus. This
observation then suggests various generalizations of these examples. A special
umbral representation of the canonical commutation relations given in terms of
the position and momentum operator on a lattice is investigated in detail.Comment: 19 pages, Late
Discrete coherent and squeezed states of many-qudit systems
We consider the phase space for a system of identical qudits (each one of
dimension , with a primer number) as a grid of
points and use the finite field to label the corresponding axes.
The associated displacement operators permit to define -parametrized
quasidistribution functions in this grid, with properties analogous to their
continuous counterparts. These displacements allow also for the construction of
finite coherent states, once a fiducial state is fixed. We take this reference
as one eigenstate of the discrete Fourier transform and study the factorization
properties of the resulting coherent states. We extend these ideas to include
discrete squeezed states, and show their intriguing relation with entangled
states between different qudits.Comment: 11 pages, 3 eps figures. Submitted for publicatio
Degree of explanation
Partial explanations are everywhere. That is, explanations citing causes that explain some but not all of an effect are ubiquitous across science, and these in turn rely on the notion of degree of explanation. I argue that current accounts are seriously deficient. In particular, they do not incorporate adequately the way in which a cause’s explanatory importance varies with choice of explanandum. Using influential recent contrastive theories, I develop quantitative definitions that remedy this lacuna, and relate it to existing measures of degree of causation. Among other things, this reveals the precise role here of chance, as well as bearing on the relation between causal explanation and causation itself
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