|
Judy A. Kennedy, James A. Yorke
|
|
Generalized Henon difference equations with delay
|
|
pp. 9-28
|
PDF version
|
|
Charles Conley once said his goal was to reveal the discrete in the continuous.
The idea here of using discrete cohomology to elicit the behavior of continuous
dynamical systems was central to his program. We combine this idea with our idea
of "expanders" to investigate a difference equation of the form
xn=F(xn-1,../../../badania/actam. ,xn-m) when F has a
special form. Recall that the equation xn=q(xn-1)
is chaotic for continuous real-valued q that satisfies q(0)<0, q(1/2)>1,
and q(1)<0 . For such a q, it is also easy to analyze
xn=q(xn-k) where k>1. But when a small
perturbation g(xn-1,../../../badania/actam. xn-m) is added, the equation
xn=q(xn-k)+g(xn-1,../../../badania/actam. ,xn-m)
(where 1 < k < m) is far harder to analyze and appears to require degree
theory of some sort. We use k-dimensional cohomology to show that this
equation has a 2-shift in the dynamics when g is sufficiently small.
|
|
Judy Kennedy
|
|
More on the shift dynamics-indecomposable continua connection
|
|
pp. 29-47
|
PDF version
|
|
If X is a compact, locally connected metric space, f:X --> X
is a homeomorphism, and Q is a closed neighborhood of X, then
Z={p\in Q:fn(p)\in Q for all integers n} is the permanent
set for f on Q, and E={p\in Q: there is some positive integer
Np such that if n>=Np, then f-n(p)\in Q}
is the entrainment set. In a previous paper, we began a study of the entrainment
sets of topological horseshoes, and showed that, under mild conditions, the closure
of the entrainment set for a topological horseshoe is "indecomposable-like" in that
it admits a continuous map onto an indecomposable continuum. Furthermore, if f
denotes the map associated with the topological horseshoe and K denotes the
closure of the entrainment set for the horseshoe, then there is a map \tilde{f}
on the indecomposable continuum, denoted \tilde{K}, and a map
h:K --> \tilde{K} such that h° \tilde{f} = f° h, i.e., the
dynamics of f on K factors over the dynamics of \tilde{f} on
\tilde{K}. Here we continue this study of the structure of entrainment sets of
topological horseshoes and investigate the presence of invariant
indecomposable continua contained in the closure of entrainment sets.
|
|
Alex Clark
|
|
The Classification of Tiling Space Flows
|
|
pp. 49-55
|
PDF version
|
|
We consider the conjugacy of the natural flows on one-dimensional tiling
spaces presented as inverse limits. We also draw connections between
geometric models and the spectral information for such flows.
|
|
Hans-Otto Walther
|
|
Differentiable semiflows for differential equations with state-dependent delays
|
|
pp. 57-66
|
PDF version
|
|
Ian Stewart
|
|
Speciation: a Case Study in Symmetric Bifurcation Theory
|
|
pp. 67-88
|
PDF version
|
|
Symmetric bifurcation theory is the study of how the trajectories of
symmetric vector fields behave as parameters are varied. We
introduce some of the basic ideas of this theory in the context of
dynamical system models of speciation in evolution. Abstractly,
these models are dynamical systems that are equivariant under the
natural permutation action of the symmetric group SN on RkN
for some integers N, k >= 1. The general theory, which is group-theoretic in
nature, makes it possible to analyse such systems in a systematic
manner. The results explain several phenomena that can be observed
in simulations of specific equations. In particular, in steady-state
bifurcation, primary branches involve bifurcation to two-species
states; such bifurcations are generically jumps; and the weighted mean
phenotype of the organisms changes smoothly, whereas the standard
deviation jumps. In particular, classical mean-field genetics, which focusses
on allele proportions in the population, cannot detect this kind of
speciation event.
|
|
Piotr Zgliczyński
|
|
Trapping regions and an ODE-type proof of the existence and uniqueness theorem
for Navier-Stokes equations with periodic boundary conditions on the plane
|
|
pp. 89-113
|
PDF version
|
|
We present a new ODE-type method of passing to the limit with the
dimension of Galerkin projection for dissipative PDEs. We apply
this method to trapping regions derived by Mattingly and Sinai to
give a new proof of the existence and uniqueness of solutions to
Navier-Stokes equations with periodic boundary conditions on the plane.
|
|
Wacław Marzantowicz
|
|
Homotopical dynamics
|
|
pp. 115-134
|
PDF version
|
|
In this paper we give a review of some recent results of study the
dynamics of a map f with use of topological invariants. We
restrict our consideration to these invariants which are
determined by the homotopy class of f. Our main object of
interest is the set of homotopy minimal periods of f, i.e.,
the set of natural numbers which are minimal periods for all maps
which are homotopic to f. We also show that the same tools
are useful in the study of minimal periods of a map of the sphere
which commutes with a free homeomorphism and in establishing that
the logarithm of spectral radius of a map of compact nilmanifold
is a lower bound for the topological entropy.
|
|
Grzegorz Graff, Piotr Nowak-Przygodzki
|
|
Sequences of fixed point indices of iterations in dimension 2
|
|
pp. 135-140
|
PDF version
|
|
Let ind(f,0) be the local fixed point index at 0. We show
that every sequence of integers which satisfies Dold relations
can be realized as {ind(fn,0)}n=1infty,
where f is a continuous self-map of a 2-dimensional disk D2.
|
|
Grzegorz Gabor
|
|
On the generalized retract method for differential inclusions with constraints
|
|
pp. 141-156
|
PDF version
|
|
In the paper, we study the problem of existence of solutions to
differential inclusions remaining in prescribed closed subsets of
a Euclidean space. We find some new homological and homotopical
sufficient conditions for existence of such trajectories. Strong
deformations and multivalued admissible deformations are used as
main tools.
|
|
Zdzisław Dzedzej
|
|
Remarks on bounded solutions for some nonautonomous ODE
|
|
pp. 157-162
|
PDF version
|
|
A Borsuk-Ulam type argument is used in order to prove existence of
nontrivial bounded solutions to some nonautonomous linear differential
equations.
|
|
Leszek Pieniążek, Klaudiusz Wójcik
|
|
Complicated dynamics in nonautonomous ODEs
|
|
pp. 163-179
|
PDF version
|
|
We present a topological method for detecting complicated dynamics
in nonautonomous ordinary differential equations (not necesserily
periodic with respect to the time variable). Our main result gives
a sufficient condition for the existence of a class of solutions,
whose presence displays some chaotic features of the dynamics. The
method is based on the Ważewski Retract Theorem and the
Lefschetz Fixed Point Theorem. Some applications to the
nonautonomous systems in the plane are considered.
|
|
Krzysztof Ciesielski
|
|
On negative escape time in semidynamical systems
|
|
pp. 181-188
|
PDF version
|
|
We present the correction of some incorrectness in the paper [12].
|
|
Anna Bistroń
|
|
On minimal and invariant sets in semidynamical systems
|
|
pp. 189-204
|
PDF version
|
|
We investigate the structure of non-trivial, weakly minimal and
negatively strongly invariant sets in a semidynamical system on a
locally compact metric space. For a negative prolongational limit
set for a semidynamical system we present two different
definition which appear to be equivalent. Certain properties of
these sets are discussed.
|
|
W. Pleśniak
|
|
Pluriregularity in polynomially bounded o-minimal structures
|
|
pp. 205-214
|
PDF version
|
|
Given a polynomially bounded o-minimal structure \goth S and a set
A\subset Rn belonging to \goth S, we show that A
(considered as a subset of Cn) is pluriregular at every point
a\in \overline{int}A that can be attained by a Cinfty arc
\gamma: [0,\epsilon] --> Rn belonging to \goth S,
such that \gamma(0)=a and \gamma((0,\epsilon])\subset int A.
In particular, if \goth S is a recently found in [22] polynomially bounded
o-minimal structure of quasianalytic functions in the sense of Denjoy-Carleman,
then any set A\subset Rn that belongs to \goth S
is pluriregular at every point a\in\overline{int}A.
|
|
Alicja Skiba
|
|
Bernstein quasianalytic functions on algebraic sets
|
|
pp. 215-223
|
PDF version
|
|
We extend the notion of Bernstein
quasianalytic functions to algebraic sets in Cn.
We prove a uniqueness principle for such functions.
|
|
Keiko Fujita
|
|
Harmonic Bergman kernel for some balls
|
|
pp. 225-234
|
PDF version
|
|
We treat the complex harmonic function on the Np-ball which
is defined by the Np-norm related to the Lie norm.
As a subspace, we treat Hardy spaces and consider the Bergman kernel
on those spaces.
Then, we try to construct the Bergman kernel in a concrete form
in 2-dimensional Euclidean space.
|
|
Anna Maria Pelczar
|
|
Some version of Gowers' dichotomy for Banach spaces
|
|
pp. 235-243
|
PDF version
|
|
In this paper another version of Gowers' dichotomy for Banach spaces,
involving topologies of special type on the Cartesian product of Banach spaces,
is presented. These topologies are closely related to the game used by W.T.Gowers
in his proof of the dichotomy.
|
|
Katarzyna Grasela
|
|
Ultraincreasing Distributions of Exponential Type
|
|
pp. 245-253
|
PDF version
|
|
In this paper Fourier transform images of Gevrey ultradistribution spaces are
described. It is proved that such spaces with the strong topology in regard to
natural duality are of the M* type in the sense of Silva. It is also
proved that the space of test functions of such images is a locally convex
convolution algebra of the LN* type. The received
results complete one known statement of Hörmander.
|
|
Piotr Kościelniak
|
|
Continuous and inverse shadowing for flows
|
|
pp. 255-266
|
PDF version
|
|
We define continuous and inverse shadowing for flows and prove some properties.
In particular, we will prove that an expansive flow without fixed points on a
compact metric space which is a shadowing is also a continuous shadowing and
hence an inverse shadowing (on a compact manifold without boundary).
|
|
G.S. Hall
|
|
Holonomy Theory and 4-dimensional Lorentz Manifolds
|
|
pp. 267-272
|
PDF version
|
|
This lecture describes the holonomy group for a 4-dimensional
Hausdorff, connected and simply connected manifold admitting a
Lorentz metric and shows, briefly, some applications to Einstein's
space-time of general relativity.
|
|
Włodzimierz M. Mikulski, Jiři M. Tomáš
|
|
The natural operators lifting k-projectable vector fields
to product-preserving bundle functors on k-fibered manifolds
|
|
pp. 273-282
|
PDF version
|
|
For any product-preserving bundle functor F defined on the
category k-\F\M of k-fibered manifolds, we determine all
natural operators transforming k-projectable vector fields on
Y\in Ob(k-\F\M) to vector fields on FY. We also determine all
natural affinors on FY. We prove a composition property
analogous to that concerning Weil bundles.
|
|
Ronald Brown, James S. Glazebrook
|
|
Connections, local subgroupoids, and a holonomy Lie groupoid of a line bundle gerbe
|
|
pp. 283-296
|
PDF version
|
|
Our main aim is to associate a holonomy Lie groupoid to the
connective structure of an abelian gerbe. The construction has
analogies with a procedure for the holonomy Lie groupoid of a
foliation, in using a locally Lie groupoid and a globalisation
procedure. We show that path connections and 2-holonomy on line
bundles may be formulated using the notion of a connection pair on
a double category, due to Brown-Spencer, but now formulated in
terms of double groupoids using the thin fundamental groupoids
introduced by Caetano-Mackaay-Picken. To obtain a locally Lie
groupoid to which globalisation applies, we use methods of local
subgroupoids as developed by Brown-Icen-Mucuk.
|
|
J. Szenthe
|
|
A construction of transverse submanifolds
|
|
pp. 297-306
|
PDF version
|
|
In case of Riemannian manifolds isometric actions admitting submanifolds which
intersect each orbit orthogonally have nice geometric properties which
generalize those of adjoint actions of compact semi-simple Lie groups as
given by their Cartan-Weyl theory [1], [4], [5]. In case of isometric
actions on Lorentz manifolds degenerate orbits may occur and this fact
renders the very definition of orthogonally transverse submanifolds
problematic, since orthogonality then does not imply transversality.
Furthermore, simple examples show that it would be too restrictive to require
that all orbits of an action should be intersected orthogonally by a single
submanifold as in the Riemannian case. For the above reasons, it seems
justified to reconsider the problem in more general affine settings. A
construction is proposed below which in the case of an affine action
under some assumptions yields a set of submanifolds intersecting generic orbits of
the highest dimension transversally. The results thus obtained are then applied to
isometric actions on Lorentz manifolds.
|
|
Anders Kock
|
|
First neighbourhood of the diagonal, and geometric distributions
|
|
pp. 307-318
|
PDF version
|
|
For any manifold, we describe the notion of geometric distribution on it,
in terms of its first neighbourhood of the diagonal. In these "combinatorial"
terms, we state the Frobenius Integrability Theorem, and use it to give
a combinatorial proof of the Ambrose-Singer Theorem on connections in
principal bundles.
|
|
A. Borowiec, M. Ferraris, M. Francaviglia, M. Palese
|
|
Conservation laws for non-global Lagrangians
|
|
pp. 319-331
|
PDF version
|
|
In the Lagrangian framework for symmetries and conservation laws of
field theories, we investigate globality properties of conserved
currents associated with non-global Lagrangians admitting global
Euler-Lagrange morphisms. Our approach is based on the recent
geometric formulation of the calculus of variations on finite
order jets of fibered manifolds in terms of variational sequences.
|
|
Stanisław Ewert-Krzemieniewski
|
|
Notes on extended recurrent and extended quasi-recurrent manifolds
|
|
pp. 333-340
|
PDF version
|
|
A local structure theorem for conformally flat manifolds of dimension n>4
with condition of recurrent type imposed on Riemann curvature tensor is
proved. It appears that the condition describes almost exactly the
subprojective manifolds.
|
