520 research outputs found
Cyclicity in families of circle maps
In this paper we will study families of circle maps of the form xâŠx+2Ïr+af(x)(mod2Ï) and investigate how many periodic trajectories maps from this family can have for a âtypicalâ function f provided the parameter a is small
Fractional differentiability of nowhere differentiable functions and dimensions
Weierstrass's everywhere continuous but nowhere differentiable function is
shown to be locally continuously fractionally differentiable everywhere for all
orders below the `critical order' 2-s and not so for orders between 2-s and 1,
where s, 1<s<2 is the box dimension of the graph of the function. This
observation is consolidated in the general result showing a direct connection
between local fractional differentiability and the box dimension/ local Holder
exponent. Levy index for one dimensional Levy flights is shown to be the
critical order of its characteristic function. Local fractional derivatives of
multifractal signals (non-random functions) are shown to provide the local
Holder exponent. It is argued that Local fractional derivatives provide a
powerful tool to analyze pointwise behavior of irregular signals.Comment: minor changes, 19 pages, Late
Observing the Symmetry of Attractors
We show how the symmetry of attractors of equivariant dynamical systems can
be observed by equivariant projections of the phase space. Equivariant
projections have long been used, but they can give misleading results if used
improperly and have been considered untrustworthy. We find conditions under
which an equivariant projection generically shows the correct symmetry of the
attractor.Comment: 28 page LaTeX document with 9 ps figures included. Supplementary
color figures available at http://odin.math.nau.edu/~jws
Local structure of self-affine sets
The structure of a self-similar set with open set condition does not change
under magnification. For self-affine sets the situation is completely
different. We consider planar self-affine Cantor sets E of the type studied by
Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the
horizontal axis is an interval. We show that within small square neighborhoods
of almost each point x in E, with respect to many product measures on address
space, E is well approximated by product sets of an interval and a Cantor set.
Even though E is totally disconnected, the limit sets have the product
structure with interval fibres, reminiscent to the view of attractors of
chaotic differentiable dynamical systems.Comment: 10 pages, 2 figure
- âŠ