The dimension spectrum of some dynamical systems

We analyze the dimension spectrum previously introduced and measured experimentally by Jensen, Kadanoff, and Libchaber. Using large-deviation theory, we prove, for some invariant measures of expanding Markov maps, that the Hausdorff dimensionf() of the set on which the measure has a singularity is a well-defined, concave, and regular function. In particular, we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number. We also show in these particular cases that the functionf is universal.     

The simple relation between certain dynamical systems

A simple relation between certain dynamical systems is established.     

A dynamical system with integer information dimension and fractal correlation exponent

In this paper we construct a family {T }, 0<<1/2, of exact endomorphisms of [0, 1] such that the invariant measurem ofT is equivalent to Lebesgue measure but has fractal correlation exponent =2. This shows that an almost complete dichotomy can exist between the information dimension and the correlation exponent in observable dynamical systems.Research supported by the Natural Sciences and Engineering Research Council of Canada     

Output functions and fractal dimensions in dynamical systems

We present a novel method for the calculation of the fractal dimension of boundaries in dynamical systems, which is in many cases many orders of magnitude more efficient than the uncertainty method. We call it the output function evaluation (OFE) method. We show analytically that the OFE method is much more efficient than the uncertainty method for boundaries with D<0.5, where D is the dimension of the intersection of the boundary with a one-dimensional manifold. We apply the OFE method to a scattering system, and compare it to the uncertainty method. We use the OFE method to study the behavior of the fractal dimension as the system's dynamics undergoes a topological transition.     

Measure and dimension of solenoidal attractors of one dimensional dynamical systems

Letf:MM be aC -map of the interval or the circle with non-flat critical points. A closed invariant subsetAM is called a solenoidal attractor off if it has the following structure: , where{I _k ⁽ⁿ⁾ is the cycle of intervals of periodp _n. We prove that the Lebesgue measure ofA is equal to zero and if sup(p _n+1/p_n)< then the Hausdorff dimension ofA is strictly less than 1.     

On a new first integral of certain dynamical systems

The analytical relation between the symmetries and the first integrals of certain dynamical systems is extended beyond the limits of the standard Hamiltonian framework. A connection symmetries and the ergodic property arises.     

The fractal dimension of the Lorenz attractor

Taken's box-counting algorithm for computing the fractal dimension of a strange attractor is applied to the Lorenz equation. A convergence problem is discussed, and an approximate dimension is computed.     

The fractal dimension of the inflationary universe

In cosmological scenarios of the new inflationary type, inflation never ends completely. The total volume of inflating regions grows exponentially with time, and they form a self-similar fractal of dimension slightly less than 3.     

On certain time characteristics of dynamical systems driven by noise

The moments of the transition time of non-linear dynamical systems driven by noise are introduced. It is demonstrated that the well-known recurrence formula for the moments of the first passage time (FTP) of the absorbing boundary by a Brownian particle can also be used to obtain the exact values of the moments of the transition time in non-linear dynamical systems with noise, described by a symmetric potential profile.     

The fractal dimension in computer-simulated random walks

Computer-simulated random walks of Powles and Rapaport have been analyzed in terms of a scale-dependent fractal dimension D(r). It turns out that these random walks perfectly fit this D-expression. This suggests that a theoretical derivation of this D-form should be possible in three-dimensional systems.     

Some variational formulas for Hausdorff dimension,topological entropy,and SRB entropy for hyperbolic dynamical systems

We derive several variational formulas for the topological entropy and SRB entropy of Axiom A flows on compact manifolds and for the Hausdorff dimension of basic sets for Axiom A diffeomorphisms on compact surfaces.     

The fractal dimension of the sem-picture of slip lines

Computer experiments modelling the plastic slip deformation of an idealized single crystal have shown that under certain conditions the fractal exponentD of the slip line pattern on the crystal surface may differ from the value ofD established on the picture of this surface.     

On the effects of an uncertainty on the evolution law in dynamical systems

We show that a small perturbation of the evolution law of chaotic dynamical systems is practically equivalent to the finite precision on the knowledge of the initial condition. In spite of the short predictability time t_p on a single trajectory we found that the statistical properties are not sensitive to small changes in the evolution law or initial probability distribution. This feature holds also for correlation functions at a delay larger than t_p. We discuss the roundoff effects on integrable systems.     

On the Hausdorff dimension of fractal attractors

We consider such mappingsx _n+1=F(x_n) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the Hausdorff dimension of the attractor is universally equal toD=0.538 ...