Hyperbolic Components Research Articles

Captures, matings and regluings

In parameter slices of quadratic rational functions, we identify arcs represented by matings of quadratic polynomials. These arcs are on the boundaries of hyperbolic components.

Mandelbrot-like set for some family of transcendental meromorphic functions

We consider one-parameter family of transcendental meromorphic functions of the form , , . We study the geometry and dynamics of hyperbolic functions and define the Mandelbrot-like set ℳ for this family. We also characterize the hyperbolic components and give combinatorial description of their deployment in the parameter plane.

Discontinuous 1-D Spherical Transport Solvers using Diffusion Preconditioner and Iterative Methods

Using discontinuous Galerkin finite-element methods to solve the hyperbolic components of the transport operator in the 1-D spherically symmetric case, the convergence behavior of various iterative methods are compared and evaluated. Diffusion synthetic accelerator, a preconditioner based on the diffusion approximation to the transport equation, is presented formally for this geometry for the first time. Compared with classical, finite-difference like methods (diamond difference methods), it is found that DG diffusion based preconditioners performed extremely well in resolving problems with strong scattering effects and material discontinuities.

Homeomorphisms between limbs of the Mandelbrot set

We prove that for every hyperbolic component of the Mandelbrot set, any two limbs with equal denominators are homeomorphic so that the homeomorphism preserves periods of hyperbolic components. This settles a conjecture on the Mandelbrot set that goes back to 1994.

Classification of almost toric singularities of Lagrangian foliations

The topological classification is given of almost toric singularities of integrable Hamiltonian systems with a large number of degrees of freedom, that is, of nondegenerate singularities without hyperbolic components. A descriptive geometric model is constructed, which makes it possible to perform effective calculations.Bibliography: 10 titles.

Dynamical approximation and kernels of non-escaping hyperbolic components

AbstractLet ℱnbe families of entire functions, holomorphically parameterized by a complex manifoldM. We consider those parameters inMthat correspond tonon-escaping hyperbolicfunctions, i.e. those mapsf∈ℱnfor which the postsingular setP(f) is a compact subset of the Fatou set ℱ(f) off. We prove that if ℱn→ℱ∞in the sense of a certain dynamically sensible metric, then every non-escaping hyperbolic component in the parameter space of ℱ∞is akernelof a sequence of non-escaping hyperbolic components in the parameter spaces of ℱn. Parameters belonging to such a kernel do not always correspond to hyperbolic functions in ℱ∞. Nevertheless, we show that these functions must beJ-stable. Using quasiconformal equivalences, we are able to construct many examples of families to which our results can be applied.

The fluid-structure interaction model with both control and disturbance at the interface: a game theory problem via an abstract approach

We consider an established hyperbolic–parabolic fluid-structure interaction model with control and disturbance acting at the interface between the two media. Here, the structure (modelled by the system of dynamic elasticity) is immersed in a fluid (modelled by the linearized Navier–Stokes equations). A game theory problem between control and disturbance is studied, when both act at the interface. To this end, the main mathematical difficulty that one encounters is the fact that such model fails to satisfy the ‘singular estimate’ from, say, control to state. This is a critical obstacle, as this is precisely the foundational property used in the development by Triggiani and Zhang [Min–max game theory and non-standard differential Riccati equations under singular estimates for e At B and e At G in the absence of analyticity, Set-Valued Var. Anal., 17 (2009), pp. 245–283.] for a full theory to include the solvability of the Differential Riccati equation in the study of the associated min–max game theory problem. Failure of the ‘singular estimate’ property is due to a mismatch between the parabolic and hyperbolic component of the overall coupled dynamics. Next, by introducing suitable observation or output operators, with incremental smoothness on the trajectory, it is shown that the resulting system satisfies a modified singular estimate, this time from the control to the observation space. This then allows one to adapt (and generalize) the complete min–max theory by Triggiani and Zhang to the present fluid-structure interaction model. More precisely, the approach followed is based on an abstract setting, of which the fluid-structure interaction model is a canonical illustration, which enjoys a desirable property not assumed for the abstract model.

Geometric construction and analytic representation of Julia sets of polynomial maps

We show that in the hyperbolic components of the Mandelbrot set, the Julia set J of a quadratic map is given by transseries formulae, rapidly convergent as repelling periodic points are approached.Up to conformal transformations, we obtain J from a smoother curve of lower Hausdorff dimension, by replacing pieces of the more regular curve by increasingly rescaled elementary ‘bricks’ obtained from the transseries expression. Self-similarity of J, up to rescalings and conformal transformations in a neighbourhood of J, is manifest in the formulae.The Hausdorff dimension of J is estimated by the transseries formula. We discuss how the analysis extends to more general polynomial maps.

A note on the jordan decomposition

The multiplicative Jordan decomposition of a linear isomorphism of Rn into its elliptic, hyperbolic and unipotent components is well know. One can define an abstract Jordan decomposition of an element of a Lie group by taking the Jordan decomposition of its adjoint map. For real algebraic Lie groups, some results of Mostow implies that the usual multiplicative Jordan decomposition coincides with the abstract Jordan decomposition. Here, for a semisimple linear Lie group, we obtain this fact by elementary methods. We also obtain the corresponding results for semisimple linear Lie algebras. Complete and simple proofs of these facts are lacking in the literature, so that the main purpose of this article is to fill this gap.

Topological regluing of rational functions

Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston--Teichm\"uller theory. We will discuss a topological theory of regluing, and trace a direction in which a holomorphic theory can develop.

A general view of pseudoharmonics and pseudoantiharmonics to calculate external arguments of Douady and Hubbard

Harmonics give us a compact formula and a powerful tool in order to calculate the external arguments of the last appearance hyperbolic components and Misiurewicz points of the Mandelbrot set in some particular cases. Antiharmonics seem however to have no application. In this paper, we give a general view of pseudoharmonics and pseudoantiharmonics, as a generalization of harmonics and antiharmonics. Pseudoharmonics turn out to be a more powerful tool than harmonics since they allow the calculation of external arguments of the Mandelbrot set in many more cases. Likewise, unlike antiharmonics, pseudoantiharmonics turn out to be a powerful tool to calculate external arguments of the Mandelbrot set in some cases.

Multipliers of periodic orbits of quadratic polynomials and the parameter plane

We prove a result about an extension of the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4 n /p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected.

Time-resolved optical spectroscopy of CsI(Tl) crystals by pulsed electron beam irradiation

Properties of the color and emission centers induced with an electron pulse beam at temperature within 80–300 K have been studied in CsI(Tl) crystals. It has been established by optical spectrometry with time resolution that initial color centers in this crystal are only Tl 0 and V k centers, which spontaneously recombine emitting visible light at 2.25 and 2.55 eV. It has been shown that the emission decay kinetics at 80 K include two fast exponential components with decay constants 3 and 14 μs as well as slow hyperbolic component with the power index depending on the wavelength of the emitting light. The temperature effect on the emission kinetics has been studied and it has been directly proved that the emission rise stage at the temperature above 170 K is caused by the recombination of electrons, which are thermally released from single Tl 0 centers, with V kA centers. The origin of scintillations in CsI(Tl) crystal is discussed in terms of the tunnel electron transitions from ground state of Tl 0 centers to ground state of V k centers at different distances from each other.

Tessellation and Lyubich–Minsky laminations associated with quadratic maps, II: Topological structures of 3-laminations

According to an analogy to quasi-Fuchsian groups, we investigate the topological and combinatorial structures of Lyubich and Minsky’s affine and hyperbolic33-laminations associated with hyperbolic and parabolic quadratic maps.We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same33-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient33-laminations of Douady’s rabbit, the Cauliflower, andz↦z2z \mapsto z^2are described.The descriptions employ a new method oftessellationinside the filled Julia set introduced in Part I [Ergodic Theory Dynam. Systems29(2009), no. 2] that works like external rays outside the Julia set.

Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics

The chaotic ray dynamics inside dielectric cavities is described by the properties of an invariant chaotic saddle. The localization of the far-field emission in specific directions, recently observed in different experiments and wave simulations, is found to be a consequence of the filamentary pattern of the saddle's unstable manifold. For cavities with mixed phase space, the chaotic saddle is divided in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential $(tl{t}_{c})$ and the asymptotic power-law $(tg{t}_{c})$ decay of the energy inside the cavity. The alignment of the manifolds of the two components of the saddle explains why even if the energy concentration inside the cavity dramatically changes from $tl{t}_{c}$ to $tg{t}_{c}$, the far-field emission changes only slightly. Simulations in the annular billiard confirm and illustrate the main results.

Poincaré recurrences and transient chaos in systems with leaks

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.

Preface

This issue comprises a selection of papers in the general area of analysis and control of systems described by non-linear evolutionary equations, that are relevant to applications in mathematical physics. Models considered range from classical non-linear wave and heat equations to quite complex systems consisting of two or more coupled equations. In this latter case, coupling often occurs between two different types of dynamics - say, a hyperbolic component and a parabolic component - with coupling in various forms, throughout the interior of the spatial domain and/or at the interface between the two media. Illustrations include systems of non-linear thermo-elasticity; fluid structure- and acoustic-structure interactions; electro-magnetism among others. <br> Dynamical models such as these are frequently encountered in modern technological applications. In recent years, they have attracted considerable attention and many new results and developments have become available. <br> Papers collected in this volume address and present some of these advances, with particular emphasis on newly developed techniques that bear on further progress in the field. <br><br> For more information please click the “Full Text” above.

Bifurcations in the space of exponential maps

This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+\kappa$. We prove that the boundary (in $\C$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon. In fact, we prove the stronger statement that the exponential bifurcation locus is connected in $\C$, which is an analog of Douady and Hubbard's celebrated theorem that the Mandelbrot set is connected. We show furthermore that $\infty$ is not accessible through any nonhyperbolic ("queer") stable component. The main part of the argument consists of demonstrating a general "Squeezing Lemma", which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components.

An Experimental Analysis of Time-Inconsistency in Long-Run Projects

In the first part of this paper, we elicit time preferences by using the experimental and econometric methods of Benhabib-Bisin-Schotter (2006). We follow the matching task procedure on money-time pairs with real rewards. Among the model specifications we use, the one with exponential discounting and quasi hyperbolic component of present bias appears to be the best model fitting the data. Unlike Benhabib et al., the present bias in the form of a fixed cost is not supported strongly by the data. In the second part of the paper, we test the theory of behavior of time-inconsistent agents in a long run project based on the quasi hyperbolic specification of O'donoghue and Rabin (2005) and Akin (2004). The preferences elicited in the first part are used to predict the behavior of agents in the long run project and categorize them based on their types. We find that the theory captures most of the subjects' observed behavior and helps understanding their types. We also find that some of the observations are compatible with alternative models, including sign effect, the preference for improving sequences, and anticipatory utility models.

A GENERALIZED VERSION OF THE MCMULLEN DOMAIN

We study the family of complex maps given by Fλ(z) = zn + λ/zn + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.