Dynamic Rays Research Articles

Topological dynamics of cosine maps

AbstractThe set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,\,b} \colon z \mapsto ae^z+be^{-z}$ for $a,\,b\in \mathbb{C}^\ast$ , consists of a collection of injective curves, called dynamic rays. If a critical value of $C_{a,\,b}$ escapes to infinity, then some of its dynamic rays overlap pairwise and split at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map $z\mapsto \cosh(z)$ . We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of $z\mapsto \cosh(z)$ , and conclude that no two of its dynamic rays land together.

Splitting Hairs with Transcendental Entire Functions

Abstract In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of their topological dynamics in terms of a simpler model has been given inspired by methods from polynomial dynamics. In this paper, and for the 1st time, we give analogous results in cases when the postsingular set is unbounded. More specifically, we show that if $f$ is of finite order, has bounded criticality on its Julia set $J(f)$, and its singular set consists of finitely many critical values that escape to infinity and satisfy a certain separation condition, then $J(f)$ is a collection of dynamic rays or hairs, which split at critical points, together with their corresponding landing points. In fact, our result holds for a much larger class of functions with bounded singular set. Moreover, this result is a consequence of a significantly more general one: we provide a topological model for the action of $f$ on its Julia set.

Singular values and non-repelling cycles for entire transcendental maps

Let $f$ be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When $f$ has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality. Our approach is combinatorial in the spirit of the approach used by [Ki00], [BCL+16] for polynomials.

About Rays, Dreadlocks and Periodic Points in Transcendental Dynamics

We study some aspects of the iteration of an entire map f over the complex plane ℂ. In many settings in complex dynamics one can define periodic curves (called dynamic rays) in the dynamical plane and study their relation with periodic points. The most famous example of this kind of results is the Douady-Hubbard landing theorem for polynomial dynamics. We describe an analogous statements for transcendental maps which satisfy some growth conditions and a further generalization to general transcendental maps with bounded postsingular set, without any growth assumption. We also describe some implications for rigidity. The results described here are from a joint work with Lasse Rempe-Gillen.

Hausdorff dimension of biaccessible angles for quadratic polynomials

A point $z$ in the Julia set of a polynomial $p$ is called biaccessible if two dynamic rays land at $z$; a point $z$ in the Mandelbrot set is called biaccessible if two parameter rays land at $z$. In both cases, we say that the external angles of these two rays are biaccessible as well. In this paper we give upper and lower bounds for the Hausdorff dimension of biaccessible external angles of quadratic polynomials, both in the dynamical and parameter space. In particular, explicitly describe those quadratic polynomials where this dimension equals 1 (if and only if the Julia set is an interval), and when it equals 0, namely, at finite direct bifurcations from the polynomial $z^2$, as well as limit points thereof.

Dynamic rays of bounded-type transcendental self-maps of the punctured plane

We study the escaping set of functions in the class $\mathcal B^*$, that is, holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is contained in a compact annulus of $\mathbb C^*$. For functions in the class $\mathcal B^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb C^*$ (and hence, in the class $\mathcal B^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every essential itinerary $e\in\{0,\infty\}^\mathbb N$, we show that the escaping set of $f$ contains a Cantor bouquet of curves that accumulate to $\{0,\infty\}$ according to $e$ under iteration by $f$.

Simple dynamic energy core equivalent rays method to design freeform surface for extended source

A simple method is proposed to design freeform surface for Lambertian extended source. In this method, it can take advantage of the designing method for point source via substituting each incident ray with a dynamically calculated equivalent ray. For each facet on the freeform surface, the equivalent ray emits from the energy weighted average-emitting-position for the corresponding incident beam, and redirects into the direction which is determined by a source-to-target mapping. The results of the designing examples show that the light distributions’ uniformities can be improved by this method, e.g., even the improvement of 59% can be achieved.

Expansivity properties and rigidity for non-recurrent exponential maps

We show that an exponential map fc(z) = ez + c whose singular value c is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its periodic dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove that f is expanding on the postsingular set with respect to the Euclidean metric in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if c is in the Julia set.

A separation theorem for entire transcendental maps

We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their landing points, separate the plane into finitely many regions, each containing exactly one interior $p-$periodic point or one parabolic immediate basin invariant under $f^p$. This result generalizes the Goldberg-Milnor Separation Theorem for polynomials, and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel discs; that "hidden components" of a bounded Siegel disc have to be either wandering domains or preperiodic to the Siegel disc itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.

A landing theorem for periodic dynamic rays for transcendental entire maps with bounded post-singular set

In this article, we present a landing theorem for periodic dynamic rays for transcendental entire maps which have bounded post-singular sets, by using standard hyperbolic geometry results.

The optical manifestation of dispersive field‐aligned bursts in auroral breakup arcs

High‐resolution optical observations of a substorm expansion show dynamic auroral rays with surges of luminosity traveling up the magnetic field lines. Observed in ground‐based imagers, this phenomenon has been termed auroral flames, whereas the rocket signatures of the corresponding energy dispersions are more commonly known as field‐aligned bursts. In this paper, observations of auroral flames obtained at 50 frames/s with a scientific‐grade Complementary Metal Oxide Semiconductor (CMOS) sensor (30° × 30° field of view, 30 m resolution at 120 km) are used to provide insight into the nature of the precipitating electrons similar to high‐resolution particle detectors. Thanks to the large field of view and high spatial resolution of this system, it is possible to obtain a first‐order estimate of the temporal evolution in altitude of the volume emission rate from a single sensor. The measured volume emission rates are compared with the sum of modeled eigenprofiles obtained for a finite set of electron beams with varying energy provided by the TRANSCAR auroral flux tube model. The energy dispersion signatures within each auroral ray can be analyzed in detail during a fraction of a second. The evolution of energy and flux of the precipitation shows precipitation spanning over a large range of energies, with the characteristic energy dropping from 2.1 keV to 0.87 keV over 0.2 s. Oscillations at 2.4 Hz in the magnetic zenith correspond to the period of the auroral flames, and the acceleration is believed to be due to Alfvenic wave interaction with electrons above the ionosphere.

Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds

Let f be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set J(f) and P(f) is finite. Assume that no asymptotic value of f belongs to J(f) and that the local degree of f at all points in J(f) is bounded by some finite constant. We prove that there is a hyperbolic map g (of the form g(z)=f(bz) for some complex number b) with connected Fatou set such that f and g are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricted to the escaping set I(g) of g. In the case where f can be written as a finite composition of maps of finite order, our theorem, together with recent results on Julia sets of hyperbolic maps, implies that J(f) is a pinched Cantor bouquet, consisting of dynamic rays and their endpoints. Our result also seems to give the first complete description of topological dynamics of an entire transcendental map whose Julia set is the whole complex plane.

Dynamic rays of bounded-type entire functions

We construct an entire function in the Eremenko-Lyubich class $\B$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\B$, in particular those of finite order, every escaping point can be connected to $\infty$ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

Dynamic rays of bounded-type entire functions

We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in B, in particular those of nite order, every escaping point can be connected to 1 by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

A landing theorem for dynamic rays of geometrically finite entire functions

A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then the Fatou set of f is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z_0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective curve consisting of escaping points of f that connects z_0 to infinity. (This curve is called a dynamic ray.) In fact, the assumption of finite order can be weakened considerably; for example, it is sufficient to assume that f can be written as a finite composition of finite-order functions.

On the structure of abstract Hubbard trees and the space of abstract kneading sequences of degree two

One of the fundamental properties of the Mandelbrot set is that the set of postcritically finite parameters is structured like a tree. We extend this result to the set of quadratic kneading sequences and show that this space contains no irrational decorations. Along the way, we prove a combinatorial analogue to the correspondence principle of dynamic and parameter rays. Our key tool is to work simultaneously with the two equivalent combinatorial concepts of Hubbard trees and kneading sequences.

Topological dynamics of exponential maps on their escaping sets

For the family of exponential maps are conjugate on suitable subsets of their escaping sets, and this conjugacy is quasiconformal. Furthermore, we prove that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's ‘pinched disk model’ for the Julia sets of these maps. On the other hand, we show that two exponential maps are generally not conjugate on their sets of escaping points. We also answer several questions about escaping endpoints of dynamic rays. In particular, we give a necessary and sufficient condition for the ray to be continuously differentiable in such a point, and show that escaping points can escape arbitrarily slowly. Furthermore, we show that the principle of topological renormalization is false for attracting exponential maps.

A landing theorem for periodic rays of exponential maps

For the family of exponential maps z ↦ exp ⁡ ( z ) + κ z\mapsto \exp (z)+\kappa , we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that g g is a periodic dynamic ray of an exponential map with nonescaping singular value. Then g g lands at a repelling or parabolic periodic point. We also show that there are periodic dynamic rays landing at all periodic points of such an exponential map, with the exception of at most one periodic orbit.

Interferometric radar observations of filamented structures due to plasma instabilities and their relation to dynamic auroral rays

Abstract. Several explanations have been proposed for Naturally Enhanced ion-acoustic Echoes observed at mid- and high-latitude Incoherent Scatter observatories. A decisive measure for distinguishing between these explanations is whether or not simultaneously observed up- and down-shifted enhancement occur simultaneously, or if they are the result of temporal and/or spatial averaging. The EISCAT Svalbard Radar has two antennas in the same radar system, which can be used as an interferometer when pointed parallel. In observations from 17 January 2002, between 06:46:10 and 06:46:30 UT, we used this possibility, in combination with direct sampling of the received signals, to yield measurements of "naturally enhanced ion-acoustic echoes" with sufficiently high resolution to resolve such averaging, if any. For the first time, radar interferometry has been employed to estimate the sizes of coherent structures. The observations were coordinated with an image intensified video camera with a narrow field of view. Together, this forms the initial study on the causal relationships between enhanced echoes and fine structure in the auroral activity on sub-kilometer, sub-second scales. The results confirm that the enhanced echoes originate from very localised regions (~300m perpendicular to the magnetic field at 500km altitude) with varying range distribution, and with high time variability (≈200ms). The corresponding increase in scattering cross section, up to 50dB above incoherent scattering, eliminates theoretical explanations based on marginal stability. The simultaneously observed up- and down-shifted enhanced shoulders, when caused by sufficiently narrow structures to be detected by the interferometer technique, originate predominantly from the same volume. These results have significant impact on theories attempting to explain the enhancements, in particular it is found that the ion-electron two-stream mechanism favoured by many authors is an unlikely candidate to explain the observations. The video data has helped establish a clear correlation between the enhanced echoes and auroral activity, on sub-second time scales, showing a threshold connection between the auroral intensity and the triggering of the radar enhancements. It appears that the up- and down-shifted enhanced echoes correlate with fine auroral structures in different ways. Key words. Ionosphere (auroral ionosphere; plasma waves and instabilities) – Radio science (interferometry)

Bifurcations of dynamic rays in complex polynomials of degree two

AbstractIn the study of bifurcations of the family of degree-two complex polynomials, attention has been given mainly to parameter values within the Mandelbrot set M (e.g., connectedness of the Julia set and period doubling). The reason for this is that outside M, the Julia set is at all times a hyperbolic Cantor set. In this paper weconsider precisely this, values of the parameter in the complement of M. We find bifurcations occurring not on the Julia set itself but on the dynamic rays landing on itfrom infinity. As the parameter crosses the external rays of M, in the dynamic plane the points of the Julia set gain and lose dynamic rays. We describe these bifurcations with the aid of a family of circle maps and we study in detail the case of the fixed points.