14,397 research outputs found
Variations of Hausdorff Dimension in the Exponential Family
In this paper we deal with the following family of exponential maps
. Denoting
the hyperbolic dimension of . It is known that the
function is real analytic in , and
that it is continuous in . In this paper we prove that this map is
C on , with . Moreover, depending on the value of
, we give estimates of the speed of convergence towards 0.Comment: 32 pages. A para\^itre dans Annales Academi{\ae} Scientiarum
Fennic{\ae} Mathematic
Fractals and dynamical chaos in a random 2D Lorentz gas with sinks
Two-dimensional random Lorentz gases with absorbing traps are considered in
which a moving point particle undergoes elastic collisions on hard disks and
annihilates when reaching a trap. In systems of finite spatial extension, the
asymptotic decay of the survival probability is exponential and characterized
by an escape rate, which can be related to the average positive Lyapunov
exponent and to the dimension of the fractal repeller of the system. For
infinite systems, the survival probability obeys a stretched exponential law of
the form P(c,t)~exp(-Ct^{1/2}). The transition between the two regimes is
studied and we show that, for a given trap density, the non-integer dimension
of the fractal repeller increases with the system size to finally reach the
integer dimension of the phase space. Nevertheless, the repeller remains
fractal. We determine the special scaling properties of this fractal.Comment: 40 pages, 10 figures, preprint for Physica
On the Hausdorff volume in sub-Riemannian geometry
For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative
of the spherical Hausdorff measure with respect to a smooth volume. We prove
that this is the volume of the unit ball in the nilpotent approximation and it
is always a continuous function. We then prove that up to dimension 4 it is
smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4
on every smooth curve) but in general not C^5. These results answer to a
question addressed by Montgomery about the relation between two intrinsic
volumes that can be defined in a sub-Riemannian manifold, namely the Popp and
the Hausdorff volume. If the nilpotent approximation depends on the point (that
may happen starting from dimension 5), then they are not proportional, in
general.Comment: Accepted on Calculus and Variations and PD
On the Hausdorff dimension of the Rauzy gasket
In this paper, we prove that the Hausdorff dimension of the Rauzy gasket is
less than 2. By this result, we answer a question addressed by Pierre Arnoux.
Also, this question is a very particular case of the conjecture stated by S.P.
Novikov and A. Ya. Maltsev in 2003.Comment: 23 pages, 5 figure
On the Numerical Study of the Complexity and Fractal Dimension of CMB Anisotropies
We consider the problem of numerical computation of the Kolmogorov complexity
and the fractal dimension of the anisotropy spots of Cosmic Microwave
Background (CMB) radiation. Namely, we describe an algorithm of estimation of
the complexity of spots given by certain pixel configuration on a grid and
represent the results of computations for a series of structures of different
complexity. Thus, we demonstrate the calculability of such an abstract
descriptor as the Kolmogorov complexity for CMB digitized maps. The correlation
of complexity of the anisotropy spots with their fractal dimension is revealed
as well. This technique can be especially important while analyzing the data of
the forthcoming space experiments.Comment: LATEX, 3 figure
- …