10,794 research outputs found
Piecewise contractions defined by iterated function systems
Let be Lipschitz contractions. Let
, and . We prove that for Lebesgue almost every
satisfying , the piecewise
contraction defined by is
asymptotically periodic. More precisely, has at least one and at most
periodic orbits and the -limit set is a periodic orbit of
for every .Comment: 16 pages, two figure
Asymptotically periodic piecewise contractions of the interval
We consider the iterates of a generic injective piecewise contraction of the
interval defined by a finite family of contractions. Let , , be -diffeomorphisms with whose images are
pairwise disjoint. Let and let be a
partition of the interval into subintervals having interior
, where and . Let be the
map given by if , for . Among other
results we prove that for Lebesgue almost every , the
piecewise contraction is asymptotically periodic.Comment: 8 page
Physical evolution in Loop Quantum Cosmology: The example of vacuum Bianchi I
We use the vacuum Bianchi I model as an example to investigate the concept of
physical evolution in Loop Quantum Cosmology (LQC) in the absence of the
massless scalar field which has been used so far in the literature as an
internal time. In order to retrieve the system dynamics when no such a suitable
clock field is present, we explore different constructions of families of
unitarily related partial observables. These observables are parameterized,
respectively, by: (i) one of the components of the densitized triad, and (ii)
its conjugate momentum; each of them playing the role of an evolution
parameter. Exploiting the properties of the considered example, we investigate
in detail the domains of applicability of each construction. In both cases the
observables possess a neat physical interpretation only in an approximate
sense. However, whereas in case (i) such interpretation is reasonably accurate
only for a portion of the evolution of the universe, in case (ii) it remains so
during all the evolution (at least in the physically interesting cases). The
constructed families of observables are next used to describe the evolution of
the Bianchi I universe. The performed analysis confirms the robustness of the
bounces, also in absence of matter fields, as well as the preservation of the
semiclassicality through them. The concept of evolution studied here and the
presented construction of observables are applicable to a wide class of models
in LQC, including quantizations of the Bianchi I model obtained with other
prescriptions for the improved dynamics.Comment: RevTex4, 22 pages, 4 figure
Hybrid Quantum Cosmology: Combining Loop and Fock Quantizations
As a necessary step towards the extraction of realistic results from Loop
Quantum Cosmology, we analyze the physical consequences of including
inhomogeneities. We consider in detail the quantization of a gravitational
model in vacuo which possesses local degrees of freedom, namely, the linearly
polarized Gowdy cosmologies with the spatial topology of a three-torus. We
carry out a hybrid quantization which combines loop and Fock techniques. We
discuss the main aspects and results of this hybrid quantization, which include
the resolution of the cosmological singularity, the polymeric quantization of
the internal time, a rigorous definition of the quantum constraints and the
construction of their solutions, the Hilbert structure of the physical states,
and the recovery of a conventional Fock quantization for the inhomogeneities.Comment: 24 pages, published in International Journal of Modern Physics A,
Special Issue: Proceedings of the Second Workshop on Quantum Gravity and
Noncommutative Geometry (Lisbon, Portugal
Modeling effective FRW cosmologies with perfect fluids from states of the hybrid quantum Gowdy model
We employ recently developed approximation methods in the hybrid quantization
of the Gowdy model with linear polarization and a massless scalar field
to obtain physically interesting solutions of this inhomogeneous cosmology.
More specifically, we propose approximate solutions of the quantum Gowdy model
constructed in such a way that, for the Hamiltonian constraint, they
effectively behave as those corresponding to a flat homogeneous and isotropic
universe filled with a perfect fluid, even though these quantum states are far
from being homogeneous and isotropic. We analyze how one can get different
perfect fluid effective behaviors, including the cases of dust, radiation, and
cosmological constant.Comment: Version accepted for publication in PR
Mode solutions for a Klein-Gordon field in anti-de Sitter spacetime with dynamical boundary conditions of Wentzell type
We study a real, massive Klein-Gordon field in the Poincar\'e fundamental
domain of the -dimensional anti-de Sitter (AdS) spacetime, subject to a
particular choice of dynamical boundary conditions of generalized Wentzell
type, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordon
equation, with the source term fixed by the normal derivative of the scalar
field at the boundary. This naturally defines a field in the conformal boundary
of the Poincar\'e fundamental domain of AdS. We completely solve the equations
for the bulk and boundary fields and investigate the existence of bound state
solutions, motivated by the analogous problem with Robin boundary conditions,
which are recovered as a limiting case. Finally, we argue that both Robin and
generalized Wentzell boundary conditions are distinguished in the sense that
they are invariant under the action of the isometry group of the AdS conformal
boundary, a condition which ensures in addition that the total flux of energy
across the boundary vanishes.Comment: 12 pages, 1 figure. In V3: refs. added, introduction and conclusions
expande
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