Numerical semigroups and periodic orbits for Markov interval maps

Interval maps constitute a very important class of discrete dynamical systems with a well developed theory. Our purpose in this paper is to study a particular class of interval maps for which the set of periods is a numerical semigroup.     

Periodic orbits from nonperiodic orbits on an interval

This paper demonstrates that any continuous real-valued function which has an orbit with infinitely many limit points must necessarily have periodic cycles of arbitrarily large prime period. We present an example of a function with an orbit whose limit points are exactly Z⁺.     

Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps

We prove that for continuous maps on the interval, the existence of an -cycle implies the existence of points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.

     

Weak hyperbolicity on periodic orbits for polynomials

We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n^5+ε with the period, for some ε>0, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with “small” multipliers. Somewhat surprisingly the proof is based on measure theorical considerations. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1113–1118.     

Foliated vector fields without periodic orbits

We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.     

Minimal orbits close to periodic frequencies

Let with h analytic of small norm. The problem of Arnold's diffusion consists in finding conditions on h which guarantee the existence of orbits Q of with connecting two arbitrary points of frequency space. Recently, J. N. Mather has found a sufficient condition for Arnold's diffusion; this condition is not read on h itself, but on the set of all action-minimizing orbits of . In this paper we try to characterize those action-minimizing orbits whose mean frequency is close to periodic. Received: November 26, 1996     

Monotone periodic orbits for torus homeomorphisms

Let be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector . Then has a topologically monotone periodic orbit with the same rotation vector.

     

Growth of periodic orbits of dynamical systems

V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 14–22, October–December, 1991.     

Existence of multiple periodic orbits on star-shaped hamiltonian surfaces

Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ? C²(?^2N, ?). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = {z ? ?^2N; H(z) = c} (c > o is a constant). The surface Σ is required to verify certain geometric assumptions: Σ bounds a star-shaped compact region ? and α? ? ? ? β? for some ellipsoid ? ? ?^2N, o < α < β. We exhibit a constant δ > O (depending in an explicit fashion on the lengths of the main axes of ? and one other geometrical parameter of Σ) such that if furthermore β²/α² < 1 + δ, then (HS) has at least N distinct geometric orbits on Σ. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland-Lasry and Ekeland). In proving this result we construct index theories for an S¹ -action, from which we derive abstract critical point theorems for S¹ -invariant functionals. We also derive an estimate for the minimal period of solutions to differential equatious.     

Degenerate resonances and branching of periodic orbits

In this paper we establish results on the existence of Lyapunov families of periodic orbits of reversible systems in around an equilibrium that presents a 0:1:1-resonance. The main proofs are based on a combined use of normal form theory, Lyapunov–Schmidt reduction and elements of symbolic computation. This work was supported by France–Brazil cooperation.     

Stability and bifurcations of symmetric periodic orbits

Se un problema Hamiltoniano integrabile non degenere, come si può ottenere dal problema dei due corpi in coordinate rotanti, è perturbato con una funzione potenziale simmetrica rispetto ad un asse, le proprietà di simmetria delle soluzioni consentono di semplificare la ricerca, di orbite periodiche. In tal modo si ottiene un teorema di continuazione delle orbite periodiche che fornisce più informazioni di quello classico. La funzione che descrive la simmetria delle orbite è genericamente una funzione di Morse, e le biforcazioni di orbite periodiche simmetriche possono essere descritte in termini di punti singolari e di valori critici di tale funzione. II problema ristretto dei tre corpi ed il problema del satellite in un potenziale non axisimmetrico sono trattati come esempi. Le stesse biforcazioni possono anche essere descritte come singolarità degeneri della funzione generatrice della trasformazione canonica associata all'applicazione di Poincaré, cioè al trascorrere di un periodo sinodico. In tal modo le proprietà di stabilità lineare delle orbite periodiche, la segnatura dei punti singolari della funzione generatrice e l'andamento qualitativo della funzione di simmetria appaiono correlate tra loro. Ne risulta la possibilità di predire, prima di qualsiasi esperimento numerico, non solo la struttura generica delle biforcazioni delle orbite periodiche simmetriche, ma anche la stabilità di tutte le orbite periodiche coinvolte.