Backward Orbits Research Articles

$$C^*$$ -Algebras Associated with Complex Dynamical Systems and Backward Orbit Structure

Let \(R\) be a rational function. The iterations \((R^n)_n\) of \(R\) gives a complex dynamical system on the Riemann sphere. We associate a \(C^*\)-algebra and study a relation between the \(C^*\)-algebra and the original complex dynamical system. In this short note, we recover the number of \(n\)th backward orbits counted without multiplicity starting at branched points in terms of associated \(C^*\)-algebras with gauge actions. In particular, we can partially imagine how a branched point is moved to another branched point under the iteration of \(R\). We use KMS states and a Perron–Frobenius type operator on the space of traces to show it.

The speed with which an orbit approaches a limit point

If is a continuous map of a compact metric space , and if is a sequence of positive reals converging to 0, we investigate the properties of the set . We show that is a dense subset of for every when x is a recurrent point, even though can be disjoint with the orbit of x for some . Under the assumption that f has an invariant non-atomic Borel probability measure , we prove results to the effect that (i) there is a uniform upper limit to the speed with which the orbit of each x can approach y for -almost every , (ii) if is ergodic with full support and if is the set of points having dense orbits, then for -almost every and for every there is a uniform upper limit to the speed with which the orbit of x can approach y. Next, using as a useful tool in proofs, we establish the following. If f is totally transitive and X is infinite, then there is a dense subset which is a countable union of Cantor sets such that and for any two distinct and any two distinct . If f is a transitive map enjoying a certain type of continuity in the backward direction, then f has a residual set of points with dense backward orbits.

Nonuniformly hyperbolic cocycles: admissibility and robustness

We give a relatively short proof of the robustness of nonuniformly hyperbolic cocycles in a Banach space under sufficiently small perturbations. In strong contrast to former proofs, we do not need to construct projections leading to the stable and unstable subspaces. Instead, these are obtained fairly explicitly depending only on the boundedness respectively of forward and backward orbits. A difficulty is that we need to construct from the beginning appropriate sequences of Lyapunov norms, with respect to which one can measure the boundedness of the orbits. These norms need not only to be guessed a priori but also all the computations would change if these were not appropriate, both for the original andfortheperturbedcocycles. Theproofoftherobustnessisbasedontherelation between the notions of nonuniform exponential dichotomy and of admissibility, together with nontrivial norm bounds for the expansion and contraction and for the norms of the projections. This relation allows us to construct an invertible operator from the set of bounded perturbations to the set of bounded solutions, and thus to conclude that under sufficiently small perturbations a similar operator exists for the perturbed cocycle.

Backward orbits of transitive maps

For a discrete dynamical system given by a Polish space X and a continuous selfmap f of X we investigate the set of all points which have some dense backward orbit. It is proved that if f is a topologically transitive map then is an analytic set which has the continuum power even locally. Under the additional assumption that f is nearly feebly open, the set is residual.

A New, Iterative, Synchronous-Response Algorithm for Analyzing the Morton Effect

Morton Effect problems involve the steady increase in rotor synchronous-response amplitudes due to differential heating across a fluid-film bearing that is induced by synchronous response. The present work presents a new computational algorithm for analyzing the Morton Effect. Previous approaches were based on Eigen or Nyquist analyses for stability studies and predicted an onset speed of instability. The present algorithm starts with a steady state elliptical orbit produced by the initial imbalance distribution, which is decomposed into a forward-precessing circular orbit and a backwards-precessing circular orbit. A separate (and numerically intensive) calculation based on the Reynolds equation plus the energy equation gives predictions for the temperature distributions induced by these separate orbits for a range of orbit radius-to-clearance ratios. Temperature distributions for the forward and backward orbits are calculated and added to produce the net temperature distribution due to the initial elliptic orbit. The temperature distribution is assumed to vary linearly across the bearing and produces a bent-shaft angle across the bearing following an analytical result due to Dimoragonas. This bent-shaft angle produces a synchronous rotor excitation in the form of equal and opposite moments acting at the bearing’s ends. For a rotor with an overhung section, the bend also produces a thermally induced imbalance. The response is due to: (1) the initial mechanical imbalance, (2) the bent-shaft excitation, and (3) the thermally-induced imbalance are added to produce a new elliptic orbit, and the process is repeated until a converged orbit is produced. For the work reported, no formal stability analysis is carried out on the converged orbit. The algorithm predicts synchronous response across the rotor’s speed range plus the speed where the response amplitudes becomes divergent by approaching the clearance. Predictions are presented for one example from the published literature, and elevated vibration levels are predicted well before the motion diverges. Synchronous-response amplitudes due to Morton Effect can be orders of magnitude greater than the response due only to mechanical imbalance, particularly near rotor critical speeds. For the example considered, bent-shaft-moment excitation produces significantly higher response levels than the mechanical imbalance induced by thermal bow. The impact of changes in: (1) bearing length-to-diameter ratio, (2) reduced lubricant viscosity, (3) bearing radius-to-clearance ratio and (4) overhung mass magnitude are investigated. Reducing lubricant viscosity and/or reducing the overhung mass are predicted to be the best remedies for Morton Effect problems.

Symbolic dynamics and Lyapunov exponents for Lozi maps

Building on the kneading theory for Lozi maps introduced by Yutaka Ishii, in 1997, we introduce a symbolic method to compute its largest Lyapunov exponent. We use this method to study the behavior of the largest Lyapunov exponent for the set of points whose forward and backward orbits remain bounded, and nd the maximum value that the largest Lyapunov exponent can assume. AMS (2000) subject classication. Primary: 37D45; Secondary: 37E30, 37E05. Keywords. Iteration of maps, Lozi maps, Symbolic dynamics, Lyapunov exponents. R esum

Backward iteration in strongly convex domains

We prove that a backward orbit with bounded Kobayashi step for a hyperbolic, parabolic or strongly elliptic holomorphic self-map of a bounded strongly convex C 2 domain in C d necessarily converges to a repelling or parabolic boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball of C d .

Investigations on the Rotordynamic Characteristics of a Hole-Pattern Seal Using Transient CFD and Periodic Circular Orbit Model

Numerical investigations on the rotordynamic characteristics of a typical hole-pattern seal using transient three-dimensional Reynolds-averaged Navier–Stokes (RANS) solution and the periodic circular orbit model were conducted in this work. The unsteady solutions combined with mesh deformation method were utilized to solve the three-dimensional RANS equations and obtain the transient reaction forces on a typical hole-pattern seal rotor at five different excitation frequencies. The relation between the periodic reaction forces and frequency dependent rotordynamic coefficients of the hole-pattern seal was obtained by considering the rotor with a periodic circular orbit (including forward orbit and backward orbit) of the seal center. The rotordynamic coefficients of the hole-pattern seal were then solved based on the obtained unsteady reaction forces and presented numerical method. Compared with the experimental data, the predicted rotordynamic coefficients of the hole-pattern seal are more agreeable with the experiment than that of the ISO-temperature (ISOT) bulk flow analysis and numerical approach with one-direction-shaking model. Furthermore, the unsteady leakage flow characteristics in the hole-pattern seal were also illustrated and discussed in detail.

Integer points in backward orbits

A theorem of J. Silverman states that a forward orbit of a rational map φ ( z ) on P 1 ( K ) contains finitely many S-integers in the number field K when ( φ ∘ φ ) ( z ) is not a polynomial. We state an analogous conjecture for the backward orbits using a general S-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map φ ( z ) = z d , and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for z n − β when β ≠ 0 is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for φ n ( z ) − β is bounded independently of n.

A remark on the effective Mordell Conjecture and rational pre-images under quadratic dynamical systems

Fix a rational basepoint b and a rational number c. For the quadratic dynamical system f c ( x ) = x 2 + c , it has been shown that the number of rational points in the backward orbit of b is bounded independent of the choice of rational parameter c. In this short Note we investigate the dependence of the bound on the basepoint b, assuming a strong form of the Mordell Conjecture.

On calibrated and separating sub-actions

We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Holder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If Ā denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A ≥ u ○ σ − u + Ā. We call contact locus of u with respect to A the subset of Σ where A = u ○ σ − u + Ā. A calibrated sub-action u gives the possibility to construct, for any point x e Σ, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Ω(A), the set of non-wandering points with respect to A.

Significance analysis of asteroid pairs

We have studied statistical significance of asteroid pairs residing on similar heliocentric orbits with distances (approximately the current relative encounter velocity between orbits) up to d = 36 m/s in the five-dimensional space of osculating elements. We found candidate pairs from the Hungaria zone through the entire main belt as well as outside the main belt, one among Hildas and one in the Cybele zone. We first determined probability that the candidate pairs are just coincidental couples from the background asteroid population. Those with estimated probability <0.3 were further investigated. In particular we computed synthetic proper elements for the relevant asteroids and used them to determine the three-dimensional distance of the members in candidate pairs. We consider small separation in the proper-element space as a signature of a real asteroid pair; conversely, cases with large separation in the proper-element space were rejected as spurious. Finally, we provide a list of candidate pairs that appear real, genetically related, to facilitate targeted studies, such as photometric and spectroscopic observations. As a by-product, we discovered six new compact clusters of three or more asteroids. Initial backward orbit integrations suggest that they are young families with ages <2 Myr.

Normal Forms, Quasi-invariant Manifolds, and Bifurcations of Nonlinear Difference-Algebraic Equations

We study the existence of quasi-invariant manifolds in a neighborhood of a fixed point of the difference-algebraic equation ($\Delta$AE) $F(z_n,z_{n+1})=0$, where $F:\mathbb{R}^{2m} \to \mathbb{R}^m$ is a smooth map satisfying $F(0,0)=0$. We demonstrate the existence of quasi-invariant manifolds on which one can define forward and backward orbits of the $\Delta$AE under mild assumptions on its linearization at the fixed point $z=0$. Indeed, by assuming this linearization to be a regular matrix pencil, one obtains a functional equation satisfied by invariant manifolds which can be solved using an extension of the contraction mapping to spaces that satisfy an interpolation property. If the $\Delta$AE under study is permitted to depend smoothly on a parameter, we then obtain a Neimark–Sacker bifurcation theorem as a corollary that can be deduced from the existence of a normal form for nonlinear $\Delta$AEs.

Simultaneous multifractal decompositions for the spectra of local entropies and ergodic averages

We consider different multifractal decompositions of the form K α i = { x : g i ( x ) = α i } , i = 1 , 2 , … , d , and we study the dimension spectrum corresponding to the multiparameter decomposition K α = ⋂ i = 1 d K α i , α = ( α 1 , … , α d ) . Then for an homeomorphism f : X → X and potentials φ, ψ : X → R we analyze the decompositions K α + = { x : lim n → ∞ 1 n ( S n + ( φ ) ) ( x ) = α } , K β - = { x : lim n → ∞ 1 n ( S n - ( ψ ) ) ( x ) = β } , where 1 n ( S n + ( φ ) ) , 1 n ( S n - ( ψ ) ) are ergodic averages using forward and backward orbits of f respectively. We must emphasize that the analysis, in any case, is done without requiring conditions of hyperbolicity for the dynamical system or Hölder continuity on the potentials. We illustrate with an application to galactic dynamics: a set of stars (which do not interact among them) moving in a galactic field.

The random case of Conley's theorem: III. Random semiflow case and Morse decomposition

In the first part of this paper, we generalize the results of the author (Liu 2006 Nonlinearity 19 277–91, 2007 Nonlinearity 20 1017–30) from the random flow case to the random semiflow case, i.e. we obtain the Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose the invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley (Conley 1978 Isolated Invariant sets and the Morse Index (Conf. Board Math. Sci. vol 38) (Providence, RI: American Mathematical Society)) to the random semiflow setting and gives a positive answer to an open problem put forward by Caraballo and Langa (Caraballo and Langa 2002 Stochastic Partial Differential Equations and Applications (Trento, 2002) (Lecture Notes in Pure and Applied Mathematics) vol 227 (New York: Dekker) pp 89–104).

INVERSE TOPOLOGICAL PRESSURE WITH APPLICATIONS TO HOLOMORPHIC DYNAMICS OF SEVERAL COMPLEX VARIABLES

In this paper, we introduce a few notions of inverse topological pressure [Formula: see text], defined in terms of backward orbits (prehistories) instead of forward orbits. This inverse topological pressure has some properties similar to the regular (forward) pressure but, in general, if the map is not a homeomorphism, they do not coincide. In fact, there are several ways to define inverse topological pressure; for instance, we show that the Bowen type definition coincides with the one using spanning sets. Then we consider the case of a holomorphic map [Formula: see text] which is Axiom A and such that its critical set does not intersect a particular basic set of saddle type Λ. We will prove that, under a technical condition, the Hausdorff dimension of the intersection between the local stable manifold and the basic set is equal to ts, i.e. [Formula: see text], for all points x belonging to Λ. Here ts represents the unique zero of the function t→P-(tϕs), with P- denoting the inverse topological pressure and [Formula: see text], y∈Λ. In general, [Formula: see text] will be estimated above by ts and below by [Formula: see text], where [Formula: see text] is the unique zero of the map t→P_(tϕs). As a corollary we obtain that, if the stable dimension is non-zero, then Λ must be a non-Jordan curve, and also, if f|Λ happens to be a homeomorphism (like in the examples from [13]), then the stable dimension cannot be zero.

PROPERTIES OF TOPOLOGICALLY TRANSITIVE MAPS ON THE REAL LINE

We prove that every topologically transitive map f on the real line must satisfy the following properties: (1) The set C of critical points is unbounded. (2)The set f(C) of critical values is also unbounded. (3)Apart from the empty set and the whole set, there can be at most one open invariant set. (4)With a single possible exception, for every element x the backward orbit {y∈R:fn(y)=x for some n in N} is dense in R.

Orbit Induced Journal Temperature Variation in Hydrodynamic Bearings

A method for evaluating the asymmetric heat input into the synchronously vibrating journal of a hydrodynamic bearing is presented. CFD techniques are used to analyze the dynamic flow and heat transport in the lubricant film and the heat input into the journal is obtained by orbit time averaging. The procedure is applied to a two-inlet circular bearing with backward and forward circular whirl journal orbits over a range of rotational speeds. The steady asymmetric heat input into the journal is found to have a sinusoidal component, which causes a steady surface temperature differential across the journal. The maximum temperature on the journal surface is always upstream of the minimum film point for forward whirl orbits and downstream for backward whirl orbits. The results provide a greater understanding of the nature of rotor thermal bending.

On the set of limit functions in wandering domains

In the iteration of transcendental meromorphic functions wandering domains may occur. In this article we discuss the behaviour of the sequence (fn ) in a wandering domain. We show that the set of limit functions may consist of any given finite set of constant functions if the constant function ∞ belongs to this set. Furthermore, we show that the set of limit functions can only exist of a finite number of constants if this set is contained in the backward orbit of ∞ or if its elements form an irrationally indifferent cycle which is contained in the Julia set.

Capture orbits and melnikov integrals in the planar three-body problem

The separatrix between bounded and unbounded orbits in the three-body problem is formed by the manifolds of forward and backward parabolic orbits. In an ideal problem these manifolds coincide and form the boundary between the sets of bounded and unbounded orbits. As a mass parameter increases the movement of the parabolic manifolds is approximated numerically and by Melnikov's method. The evidence indicates that for positive values of the mass parameter these manifolds no longer coincide, and that capture and oscillatory orbits exist.