Rate of escape of some chaotic Julia sets

We give a formula for the rates of escape for Julia sets with preperiodic critical points and forC endomorphisms of the interval with non-flat pre-periodic critical points outside the basin of attracting periodic points.Research supported by CNPq, Brasil     

Markov partition in non-hyperbolic interval dynamics

We considerC ² unimodal mapsf such that all periodic points are hyperbolic, the critical point is non-degenerated and non-recurrent, and the Julia set does not contain intervals. We construct a Markov partition for a big part of the Julia set. Then we use it to estimate the limit capacity and Hausdorff dimension of the Julia set.Partially supported by CNPq, S.C.T.-Brazil     

Escape dynamics in collinear atomic-like three mass point systems

The present paper studies the escape mechanism in collinear three point mass systems with small-range-repulsive/large-range-attractive pairwise interaction. Specifically, we focus on the asymptotic behaviour for systems with non-negative total energy.On the zero energy level set there are two distinct asymptotic states, called 1+1+1escape configurations, where all the three separations infinitely increase as t→∞. We show that 1+1+1 escapes are improbable by proving that the set of initial conditions leading to such asymptotic configurations has zero Lebesgue measure. When the outer mass points are of the same kind we deduce the existence of a heteroclinic orbit connecting the 1+1+1 escape configurations. We further prove that this orbit is stable under parameter perturbation.In the positive energies’ case, we show that the set of initial conditions leading to 1+1+1 escape configurations has positive Lebesgue measure.     

Fractal Dimension of Julia Set for Nonanalytic Maps

The Hausdorff dimensions of the Julia sets for nonanalytic maps f(z) = z ²+εz* and f(z)=z*²+ε are calculated perturbatively for small ε. It is shown that Ruelle's formula for the Hausdorff dimensions of analytic maps cannot be generalized to nonanalytic maps.     

On the dimension of deterministic and random Cantor-like sets,symbolic dynamics,and the Eckmann-Ruelle Conjecture

In this paper we unify and extend many of the known results on the dimension of deterministic and random Cantor-like sets in ? ⁿ , and apply these results to study some problems in dynamical systems. In particular, we verify the Eckmann-Ruelle Conjecture for equilibrium measures for Hölder continuous conformal expanding maps and conformal Axiom A^# (topologically hyperbolic) homeomorphims. We also construct a Hölder continuous Axiom A^# homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere. this example shows that the non-conformal Hölder continuous version of the Eckmann-Ruelle Conjecture is false. The Cantor-like sets we consider are defined by geometric constructions of different types. The vast majority of geometric constructions studied in the literature are generated by a finite collection ofp maps which are either contractions or similarities and are modeled by the full shift onp symbols (or at most a subshift of finite type). In this paper we consider much more general classes of geometric constructions: the placement of the basic sets at each step of the construction can be arbitrary, and they need not be disjoint. Moreover, our constructions are modeled by arbitrary symbolic dynamical systems. The importance of this is to reveal the close and nontrivial relations between the statistical mechanics (and especially the absence of phase transitions) of the symbolic dynamical system underlying the geometric construction and the dimension of its limit set. This has not been previously observed since no phase transitions can occur for subshifts of finite type. We also consider nonstationary constructions, random constructions (determined by an arbitrary ergodic stationary distribution), and combinations of the above.     

On the iteration of a rational function: Computer experiments with Newton's method

Using Newton's method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function. This article describes the behavior of Newton iteration for cubic polynomials. After a change of variables, these polynomials can be parametrized by a single complex parameter, and the Newton transformation has a single critical point other than its fixed points at the roots of the polynomial. We describe the behavior of the orbit of the free critical point as the parameter is varied. The Julia set, points where Newton's method fail to converge, is also pictured. These sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle.     

Chaotic complex dynamics and Newton's method

We consider one-parameter families of Julia sets arising from Newton's method in the complex domain. We show the existence of bifurcation points where zeros coalesce or change from attractors to repellors, and points where chaotic behavior occurs.     

复系统Julia集的同步

非线性系统中的分形集——Julia集, 在工程技术中有着十分重要的应用,定义了不同系统间的Julia集同步的概念, 并引入一种非线性耦合的方法, 对同一系统不同参数的Julia集进行了有效的同步.并以多项式形式和三角函数形式的Julia集同步为例验证了该方法的有效性.     

复系统Julia集的同步

非线性系统中的分形集——Julia集, 在工程技术中有着十分重要的应用,定义了不同系统间的Julia集同步的概念, 并引入一种非线性耦合的方法, 对同一系统不同参数的Julia集进行了有效的同步.并以多项式形式和三角函数形式的Julia集同步为例验证了该方法的有效性. 关键词： Julia集 同步 分形     

Quadratic-Like Dynamics of Cubic Polynomials

A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.     

Constructing Locally Connected Non-Computable Julia Sets

A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected. This research was partially conducted during the period the first author was employed by the Clay Mathematics Institute as a Liftoff Fellow. The second author’s research is supported by NSERC operating grant.     

Zur Lage des Maximums der Ausbeutekurve bei der Sekundärelektronenemission

There exists a quantitative connection between penetration depth of primary electrons, the maximum escape depth of secondary electrons and the position of the maximum of the yield curve if one assumes: (1) the maximum of the yield curve is determined by an energyE _p ^max of the primary electrons where the maximum penetration depth of primary electrons equals the maximum escape depth of the secondary electrons; and (2) the maximum escape depth of secondary electrons is given by 10 atomic layers for all metals investigated as it has been found earlier by Mayer and Hölzl for potassium. These assumptions are confirmed experimentally by measurements of yield curves and energy distributions at defined conditions and by variation of several parameters as temperature, contamination of the surface, surface roughness, evaporation conditions, and angle of incidence of the primary electrons.     

Physical meaning for Mandelbrot and Julia sets

We consider the dynamics of a kicked charged particle moving in a double-well potential and a time-dependent magnetic field. In certain cases the stroboscopic dynamics reduces to the complex logistic map, thus providing physical meaning for the Mandelbrot set. In other cases we obtain iterated function systems consisting of the inverse complex logistic map, thus providing physical meaning for Julia sets. Our approach can be generalized to complex mappings with a maximum of order q.     

Mellin transforms and correlation dimensions

We consider the Mellin transform of the correlation integrals and show that the divergence abscissa is the correlation dimension. The analytic structure of the Mellin transform is explicitly described for some Julia and Cantor sets. The existence of oscillations in the correlation integral for the Cantor sets is proved. Extensions of the results to the order d correlation integrals are discussed.     

Hausdorff and Packing Spectra,Large Deviations,and Free Energy for Branching Random Walks in {\mathbb{R}^d}

Consider an \({\mathbb{R}^d}\) -valued branching random walk (BRW) on a supercritical Galton Watson tree. Without any assumption on the distribution of this BRW we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets E(K) of infinite branches in the boundary of the tree (endowed with its standard metric) along which the averages of the BRW have a given closed connected set of limit points K. This goes beyond multifractal analysis, which only considers those level sets when K ranges in the set of singletons \({\{\alpha\}, \alpha \in \mathbb{R}^d}\) . We also give a 0–∞ law for the Hausdorff and packing measures of the level sets E({α}), and compute the free energy of the associated logarithmically correlated random energy model in full generality. Moreover, our results complete the previous works on multifractal analysis by including the levels α which do not belong to the range of the gradient of the free energy. This covers in particular a situation that was until now badly understood, namely the case where a first order phase transition occurs. As a consequence of our study, we can also describe the whole singularity spectrum of Mandelbrot measures, as well as the associated free energy function (or L ^q -spectrum), when a first order phase transition occurs.     

Atomic carbon in magnesium oxide single crystals—depth profiling,temperature-and time-dependent behavior

Using the ¹²Cd,p) ¹³C method and ultrahigh vacuum techniques it is shown that MgO single crystals, grown by arc fusion, invariably contain high concentrations of carbon, 250–2500 at.-ppm. The carbon sets up steep concentration gradients in the 0–1μm subsurface zone, disappears and reappears as a function of heat treatments between 300 and 1170 K in ultrahigh vacuum. By heating in O₂ the carbon can be burnt out of the subsurface zone, but upon isothermal annealing at 470 K the carbon starts to diffuse from the bulk back into the subsurface zone within minutes, giving rise to fluctuating concentration variations which lasted for hours with peroidicities of about 45 min. After quenching samples previously heated in O₂ to 80 K and reheating linearly at 2K/min the onset temperature of the carbon mobility was found to be as low as 140 K. From these experiments it is concluded that carbon is atomically dissolved in the MgO. The C atoms probably occupy extrinsic cation vacancies introduced in the MgO by co-dissolved “water”, but at the same time they must also be on interstitial sites where they acquire an extremely high mobility.     

Structure of 0+ states in 18O

Analysis of spectroscopic factors obtained in single-nucleon transfer reactions leading to and from ¹⁸O yields two different sets of wave functions for the first three O⁺ states. One set of wave functions is in agreement with ¹⁶O(t, p)¹⁸O data for the three states-the other set is not. The wave functions that agree with the experimental data have the majority of the $\text{(s}\text{1}\text{2}\text{)}{}^2$ strength in the third O⁺ state.     

Scaling invariance for the escape of particles from a periodically corrugated waveguide

The escape dynamics of a classical light ray inside a corrugated waveguide is characterised by the use of scaling arguments. The model is described via a two-dimensional nonlinear and area preserving mapping. The phase space of the mapping contains a set of periodic islands surrounded by a large chaotic sea that is confined by a set of invariant tori. When a hole is introduced in the chaotic sea, letting the ray escape, the histogram of frequency of the number of escaping particles exhibits rapid growth, reaching a maximum value at np and later decaying asymptotically to zero. The behaviour of the histogram of escape frequency is characterised using scaling arguments. The scaling formalism is widely applicable to critical phenomena and useful in characterisation of phase transitions, including transitions from limited to unlimited energy growth in two-dimensional time varying billiard problems.