Backward Orbits Research Articles

Backward Rauzy-Veech algorithm and horizontal saddle connections

The goal of this note is to study the action of the backward Rauzy-Veech algorithm on the translation surfaces with horizontal saddle connections. In particular, we prove that the orbit of a translation surface via the aforementioned algorithm is $\infty$-complete if and only if the surface does not posses horizontal saddle connections. Moreover, we show that appearance of all saddle connections as sides of the polygonal representations of the translations surface along the backward Rauzy-Veech induction orbit is equivalent to the minimality of the horizontal translation flow.

Equilibrium States of Endomorphisms of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{P}^{k}$\\end{document}: Spectral Stability and Limit Theorems

We establish the existence of a spectral gap for the transfer operator induced on \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{P}^{k} = \\mathbb{P}^{k} (\\mathbb{C})$\\end{document} by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).

Speeds of convergence for petals of semigroups of holomorphic functions

We study the backward dynamics of one-parameter semigroups of holomorphic self-maps of the unit disk. More specifically, we introduce the speeds of convergence for petals of the semigroup, namely the total, orthogonal, and tangential speeds. These are analogous to speeds of convergence introduced by Bracci, yet profoundly different due to the nature of backward dynamics. Results are extracted on the asymptotic behavior of speeds of petals, depending on the type of the petal. We further discuss the asymptotic behavior of the hyperbolic distance along non-regular backward orbits.

Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces

We study the interplay between the backward dynamics of a non-expanding self-map f of a proper geodesic Gromov hyperbolic metric space X and the boundary regular fixed points of f in the Gromov boundary as defined in [8]. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behavior of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains Ω⊂⊂Cq, where Ω is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with q=2, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc D⊂C obtained by Bracci and Poggi-Corradini in [14,27,28]. In particular, with our geometric approach we are able to answer a question, open even for the unit ball Bq⊂Cq (see [5,26]), namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary.

Cyclotomic and abelian points in backward orbits of rational functions

We prove several results on backward orbits of rational functions over number fields. First, we show that if K is a number field, ϕ∈K(x) and α∈K then the extension of K generated by the abelian points (i.e. points that generate an abelian extension of K) in the backward orbit of α is ramified only at finitely many primes. This has the immediate strong consequence that if all points in the backward orbit of α are abelian then ϕ is post-critically finite. We use this result to prove two facts: on the one hand, if ϕ∈Q(x) is a quadratic rational function not conjugate over Qab to a power or a Chebyshev map and all preimages of α are abelian, we show that ϕ is Q-conjugate to one of two specific quadratic functions, in the spirit of a recent conjecture of Andrews and Petsche. On the other hand we provide conditions on a quadratic rational function in K(x) for the backward orbit of a point α to only contain finitely many cyclotomic preimages, extending previous results of the second author. Finally, we give necessary and sufficient conditions for a triple (ϕ,K,α), where ϕ is a K-Lattès map over a number field K and α∈K, for the whole backward orbit of α to only contain abelian points.

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfuß conjecture [Garrido-Atienza, et al., (2010) [14]] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear multiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs.

Quasi-geodesics and backward orbits under semigroups of holomorphic functions

We explore two properties of backward orbits under semigroups of holomorphic self-maps in the unit disk. First, we prove that regular backward orbits are quasi-geodesics for the hyperbolic distance of the unit disk. Then we show that backward orbits satisfy a useful property, this time in Euclidean terms.

Remarks on algebraic dynamics in positive characteristic

AbstractIn this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.

Primitive prime divisors in backward orbits

Let f be a polynomial, a,b be two algebraic numbers, and (an)n≥1 be a sequence satisfying f(a1)=a and f(an)=an−1 for all n≥2. We generalize the concept of primitive prime divisors to the shifted sequence (an−b)n≥1 and study the generalized Zsigmondy set, that is, the set of n such that all prime ideal divisors of an−b divide at least one prior term. The finiteness of Zsigmondy sets are determined in various cases.

Birkhoff centre and backward limit points

We suggest one complete and one partial solution to the selected problems presented in the recently published article On Backward Attractors of Interval Maps (Hantáková and Roth (2021) [15]). Specifically we prove a conjecture proposing a characterisation of sets of β-limit points (i.e. limit points of all accumulation points of backward orbit branches of a specific point) for graph maps. We show that β-limit sets coincide with Birkhoff centre Rec(f)‾ and that the condition for a point to belong to its β-limit set is equivalent to belonging to the β-limit set of an other point.In the second part of the paper we deal with genericity of having all sα-limit sets closed and we prove that maps with not all sα-limit sets closed are dense in C0([0,1]), which partially solves an open problem also suggested in the aforementioned article.

Compact Sets in Petals and Their Backward Orbits Under Semigroups of Holomorphic Functions

Let (ϕt)t≥ 0 be a semigroup of holomorphic functions in the unit disk mathbb {D} and K a compact subset of mathbb {D}. We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk.

Two remarks on Narkiewicz’s property (P)

Due to Narkiewicz a field F has property (P) if for no polynomial fin F[x] of degree at least two there is an infinite f-invariant subset of F. We present a new example of an algebraic extension of {mathbb {Q}} satisfying (P). This is the first example in which we can find points of arbitrarily small positive Weil-height. Moreover, we study the possibility of property (P) for the field generated by all symmetric Galois extensions of {mathbb {Q}}. In particular we prove that there are no infinite backward orbits of non linear polynomials in this field.

Kinematic characteristics of the Milky Way globular clusters based on Gaia DR2 data

Using the data from Gaia (ESA) Data Release 2 we performed the orbital calculations of globular clusters (GCs) of the Milky Way. To explore possible close encounters (or collisions) between the GCs, using our own developed high-order φ-GRAPE code, we integrated backward and forward orbits of 119 objects with reliable positions and proper motions. In the calculations, we adopted a realistic axisymmetric Galactic potential (bulge + disk + halo). Using different impact conditions, we found four pairs of six GCs that may have experienced an encounter within twice the sum of the half-mass radii ('collisions') over the last 5 Gyr: Terzan 3 — NGC 6553, Terzan 3 — NGC 6218, Liller 1 — NGC 6522 and Djorg 2 — NGC 6553.

Kinematic characteristics of the Milky Way globular clusters based on Gaia DR2 data

Using the data from Gaia (ESA) Data Release 2 we performed the orbital calculations of globular clusters (GCs) of the Milky Way. To explore possible close encounters (or collisions) between the GCs, using our own developed high-order φ-GRAPE code, we integrated backward and forward orbits of 119 objects with reliable positions and proper motions. In the calculations, we adopted a realistic axisymmetric Galactic potential (bulge + disk + halo). Using different impact conditions, we found four pairs of six GCs that may have experienced an encounter within twice the sum of the half-mass radii ('collisions') over the last 5 Gyr: Terzan 3 — NGC 6553, Terzan 3 — NGC 6218, Liller 1 — NGC 6522 and Djorg 2 — NGC 6553.

On backward attractors of interval maps* *The research was supported by RVO, Czech Republic funding for IČ47813059. The first author was supported by the Marie Skłodowska-Curie Grant Agreement No. 883748 from the European Union’s Horizon 2020 research and innovation programme.

Special α-limit sets (sα-limit sets) combine together all accumulation points of all backward orbit branches of a point x under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of sα-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong’s attracting centre that completely characterizes which interval maps have all sα-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh’s models of solenoidal and basic ω-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of sα-limit sets to the dynamics within them. For example, we show that the isolated points in a sα-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the sα-limit set is nowhere dense. Moreover, we show that sα-limit sets in the interval are always both F σ and G δ . Finally, since sα-limit sets need not be closed, we propose a new notion of β-limit sets to serve as backward attractors. The β-limit set of x is the smallest closed set to which all backward orbit branches of x converge, and it coincides with the closure of the sα-limit set. At the end of the paper we suggest several new problems about backward attractors.

Permutable quasiregular maps

AbstractLet f and g be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$ . We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and $g = \phi \circ f$ , where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

Orbit Decomposition Method for Rotordynamic Coefficients Identification of Annular Seals

The elliptical orbit whirl model is widely used to identify the frequency-dependent rotordynamic coefficients of annular seals. The existing solution technique of an elliptical orbit whirl model is the transient computational fluid dynamics (CFD) method. Its computational time is very long. For rapid computation, this paper proposes the orbit decomposition method. The elliptical whirl orbit is decomposed into the forward and backward circular whirl orbits. Under small perturbation circumstances, the fluid-induced forces of the elliptical orbit model can be obtained by the linear superposition of the fluid-induced forces arising from the two decomposed circular orbit models. Due to that the fluid-induced forces of circular orbit, the model can be calculated with the steady CFD method, and the transient computations can be replaced with steady ones when calculating the elliptical orbit whirl model. The computational time is significantly reduced. To validate the present method, its rotordynamic results are compared with those of the transient CFD method and experimental data. Comparisons show that the present method can accurately calculate the rotordynamic coefficients. Elliptical orbit parameter analysis reveals that the present method is valid when the whirl amplitude is less than 20% of seal clearance. The effect of ellipticity on rotordynamic coefficients can be ignored.

How two planets likely acquired their backward orbits

It’s challenging enough to figure out what exoplanetary systems look like today. Reconstructing their histories is more difficult still.

Homoclinic Bifurcation of a Quadratic Family of Real Functions with Two Parameters

In this work the homoclinic bifurcation of the family H={h(a,b)(x)=ax2+b:a∈R/{0},b∈R} is studied. We proved that this family has a homoclinic tangency associated to x=0 of P1 for b=-2/a. Also we proved that Wu(P1) does not intersect the backward orbit of P1 for b>-2/a, but has intersection for b0. So H has this type of the bifurcation.

The role of the mass ratio in ballistic capture

ABSTRACT A massless particle can be naturally captured by a celestial body with the aid of a third body. In this work, the influence of the mass ratio on ballistic capture is investigated in the planar circular restricted three-body problem (CR3BP) model. Four typical dynamical environments with decreasing mass ratios, that is, the Pluto–Charon, Earth–Moon, Sun–Jupiter, and Saturn–Titan systems, are considered. A generalized method is introduced to derive ballistic capture orbits by starting from a set of initial conditions and integrating backward in time. Particular attention is paid to the backward escape orbits, following which a test particle can be temporarily trapped by a three-body gravity system, although the particle will eventually deviate away from the system. This approach is applied to the four candidate systems with a series of Jacobi constant levels to survey and compare the capture probability (quantitatively) and capture capability (qualitatively) when the mass ratio varies. Capture mechanisms inducing favourable ballistic capture are discussed. Moreover, the possibility and stability of capture by secondary celestial bodies are analysed. The obtained results may be useful in explaining the capture phenomena of minor bodies or in designing mission trajectories for interplanetary probes.