Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions

If f is in the Eremenko–Lyubich class B (transcendental entire functions with bounded singular set), then Ω = { z : | f ( z ) | > R } and f | Ω must satisfy certain simple topological conditions when R is sufficiently large. A model ( Ω , F ) is an open set Ω and a holomorphic function F on Ω that satisfy these same conditions. We show that any model can be approximated by an Eremenko–Lyubich function in a precise sense. In many cases, this allows the construction of functions in B with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do.

Let f be a transcendental entire function in the Eremenko–Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.

Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces an entire function of infinite order. We construct the first examples of entire functions of finite order in the class $\mathcal B$ with wandering domains. As in Bishop's example, these wandering domains are of oscillating type, that is, they have an unbounded non-escaping orbit. To construct such functions we use quasiregular interpolation instead of quasiconformal folding, which is much more straightforward. Our examples have order $p/2$ for any $p\in\mathbb{N}$ and, since the order of functions in the class $\mathcal B$ is at least $1/2$, we achieve the smallest possible order. Finally, we can modify the construction to obtain functions of finite order in the class $\mathcal B$ with any number of grand orbits of wandering domains, including infinitely many.

Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko–Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show that these bounds are also sharp in the Speiser class. We apply this method also to construct meromorphic functions in the Speiser class with preassigned dimensions of the Julia set and the escaping set.

The study of the dynamics of polynomials is now a major field of research, with many important and elegant results. The study of entire functions that are not polynomials – in other words transcendental entire functions – is somewhat less advanced, in part due to certain technical differences compared to polynomial or rational dynamics.In this paper we survey the dynamics of functions in the Eremenko-Lyubich class,B\mathcal {B}. Among transcendental entire functions, those in this class have properties that make their dynamics most “similar” to the dynamics of a polynomial, and so particularly accessible to detailed study. Many authors have worked in this field, and the dynamics of classB\mathcal {B}functions is now particularly well-understood and well-developed. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. There is also an increasing expectation that learning new results in transcendental dynamics will lead to a better understanding of the polynomial and rational cases.We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give stepping-stones to deeper inquiry.

For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which are Cantor circles, and prove that for any Cantor circle hyperbolic component <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the space of rational maps, the infimum of the Hausdorff dimensions of the Julia sets of the maps in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equal to the conformal dimension of the Julia set of any representative <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f 0 element-of script upper H"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f_0\in \mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and that the supremum of the Hausdorff dimensions is equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set J(f) are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions fK, K ∈ N, such that the box and packing dimensions of JfK are equal to two, even though as K → ∞ the Hausdorff dimension of JfK tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function. 2000 Mathematics Subject Classification 30D05, 37F10, 37F15, 37F35, 37F50.

We study new phenomena associated with the dynamics of higher dimensional non-invertible, hyperbolic maps f on basic sets of saddle type; the dynamics in this case presents important differences from the case of diffeomorphisms or expanding maps. We show that the stable dimension (i.e. the Hausdorff dimension of the intersection between local stable manifolds and the basic set) and the unstable dimension (similar definition) give a lot of information about the dynamical/ergodic properties of endomorphisms on folded basic sets. We prove a geometric flattening phenomenon associated to the stable dimension, i.e. we show that if the stable dimension is zero at a point, then the fractal Λ must be contained in a submanifold and f is expanding on Λ. We characterize folded attractors and folded repellers, as those basic sets with full unstable dimension, respectively with full stable dimension. We classify possible dynamical behaviors, and establish when is the system (Λ,f,μ) 1-sided or 2-sided Bernoulli for certain equilibrium measures μ on folded basic sets, for a class of perturbation maps.

Due to the fact that the range history of bistatic synthetic aperture radar (SAR) doesn’t follow the hyperbolic function, the accurate analytical spectrum of bistatic SAR cannot be obtained directly by the principle of stationary phase (POSP). In this paper, we approximate the range history of bistatic SAR by the hyperbolic function, and propose a novel two-dimensional (2-D) analytical spectrum model for bistatic SAR. The huge advantage of this novel 2-D analytical spectrum is that the equivalent radial velocity of any scatter point in the observed scene, with respect to the radar platform, is not only the same but also invariant with the equivalent range. Further, based on this novel spectrum model, the wavenumber domain algorithm (WDA) could be applied to image the whole observed scene. Finally, the simulation and realistic data imaging results also validate that this novel spectrum model is correct and effective.

For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

In this paper we survey some author’s results and developments relating the so-called Gel’fond-Leont’ev (G-L) operators of generalized integration and differentiation, classes of special functions (SF) of generalized hypergeometric type and the operators of generalized fractional calculus (GFC). The G-L operators have been introduced by Gel’fond-Leont’ev [9] in the classes of analytic functions in disks ΔR = {|z| < R}, by means of of multipliers’ sequences composed by the coeffcients of suitable entire (generating) functions. Introducing classes of SF related to Fractional Calculus (FC), as the Mittag-Leffer (ML) function, the multi-index Mittag-Leffer (multi-ML) function and its various particular cases ([16]–[18]), we specify the G-L operators generated by these entire functions. It is shown that in these cases, the G-L operators can be extended to analytic functions in wider complex domains Ω starlike with respect to the origin z = 0 and represented by operators of the Generalized Fractional Calculus (GFC), Kiryakova [14], i.e. operators of generalized integration and differentiation of arbitrary fractional multi-order. Illustrative examples and some open problems are proposed.

Contemporary theories of binary polymer blend interfaces incorporate such features of real polymer blends as compressibility, local correlations, monomer structure, etc. However, these theories require complicated numerical schemes, and their solutions often cannot be interpreted in a physically clear fashion. We develop a variational formalism for computing interfacial properties of binary polymer blends based on a hyperbolic tangent representation for the interfaces. While such an analysis is straightforward in the incompressible limit, the extension to compressible binary blends requires two distinct width parameters and nontrivial analysis. When the profile width parameters are chosen to minimize the excess free energy of a phase separated binary blend, then the interfacial properties computed from our simplified interfacial theory closely match those computed with the much more sophisticated (and computationally intensive) treatments. Significant attention is devoted to describing the interfacial properties of blends in the regime intermediate between the strong and the weak segregation limits as well as to extrapolating between these limits. The extension of the square gradient theory to the Tang–Freed quartic approximation provides a more precise definition of the weak segregation limit, but the treatment is found to overestimate both the interfacial tension and width in the strong segregation limit. The width parameters for the different components of a strongly asymmetric compressible blend vary to a lesser extent than an asymptotic analysis in the bulk suggests. This finding indicates that the central portion of the profile contributes the most in the minimization of the excess free energy with respect to the variational width parameters.

Although the Geiger-mode avalanche photodiode (GM-APD) flash ladar system offers the advantages of high sensitivity and simple construction, its detection performance is influenced not only by the incoming signal-to-noise ratio but also by the absolute number of noise photons. In this paper, we deduce a hyperbolic approximation to estimate the noise-photon number from the false-firing percentage in a GM-APD flash ladar system under dark conditions. By using this hyperbolic approximation function, we introduce a method to adapt the aperture to reduce the number of incoming background-noise photons. Finally, the simulation results show that the adaptive-aperture method decreases the false probability in all cases, increases the detection probability provided that the signal exceeds the noise, and decreases the average ranging error per frame.

W. Bergweiler proved in 1990 that the equation f∘g= Q has infinitely many solutions in the complex plane, provided f and g are transcendental entire functions and Q is a non-constant polynomial. We prove a related result, using standard notations from the Nevanlinna theory: Let f be a non-linear entire function of finite order of growth, let g be a transcendental entire function of finite lower order and let Q be a non-constant entire function such that T(r,Q)=S(r,g). If II denotes the canonical product formed with the zeros of f∘g Q, then T(r,φ) ≠ S(r,g). As an immediate consequencef∘g−Q has infinitely many zeros.

In this note it is proved that entire functions of the form $$ F_a \left( z \right) = H\left( z \right) - a\alpha \left( z \right) $$ , where H and α are entire functions, with α having at least one zero, H′ and α′ having no common zeros and T(r, α) = S(r, H), are prime in entire sense for most values of a. The class of these functions is seen to contain examples of both periodic and non-periodic prime entire functions found by Noda, Qiao, Urabe and by Singh and Charak. Another result due to Singh and Charak will be improved as well.