Narrow Results

Let M be an arbitrary nonempty set and for t ∈ M be continuous mappings of the unit circle . The aim of this paper is to investigate the existence of solutions (Φ, c), where is a continuous function and , of the following system of Schröder equations

We consider several discrete dynamical systems for which some invariants can be found. Our study includes complex Möbius transformations as well as the third-order Lyness recurrence.

This paper introduces the discrete distribution of ascent probabilities , generalizing the concept of rotation number ω, already being defined in one-dimensional unimodal maps. The map domain is partitioned into subintervals , each one containing orbits with their route characterized by a specific number of n successive ascents before a descent occurs. Then, P_n is defined as the probability to have an orbit performing n successive ascents, and equals the portion of the invariant measure within I_n. The rotation number is found to be equal to ω = P₀ = P₁, that is the portion of invariant measure within I₀ or I₁. Some significant applications of this relation concern (i) the rotation number dependence on the nonlinear parameter p, (ii) the analytical derivation of the rotation number, given the invariant density explicit expression, (iii) an easy computation of the rotation number that characterizes a periodic window. Moreover, the dependence of the ascent probabilities on the nonlinear parameter p is examined. Emphasis is placed on the discrete distribution of the ascent probabilities within the chaotic zone. A particular set of nonlinear parameter values is affiliated to the concept of ascent probabilities: For each probability P_n, n = 0, 1, 2, …, there is a lower limit of the nonlinear parameter values, p_(n+1), so that, P_n = 0 ∀ p ≤ p_(n+1). The set is analytically determined, and its specific arrangement in the chaotic zone is studied. Finally, the "u-S-P equivalence" between the triplet of the set of the fixed point and its preimages, , of the invariant density S, and of the set of the ascents probabilities, , is formulated. In particular, we show that each component of this triplet can be estimated whenever the other two components are given. Applications in the case of Logistic map are thoroughly examined.

The van der Pol circuit with neon glow lamp operates as a relaxation oscillator driven by a sinusoidal voltage source. Its dynamics is described by the function which maps an arc of circle into itself, preserves orientation on this arc, and has at most one jump-discontinuity point on the circle. Properties of this mapping are discussed with emphasis on rotation numbers, types of periodic orbits, saddle-node and border-collision bifurcations. The so-called period-adding phenomenon is explained. Some remarks are given about dynamics, where the mapping does not preserve the orientation because of too long neon lamp lighting.

We study the existence of periodic solutions of the nonautonomous periodic Lyness' recurrenceu_n+2 = (a_n + u_n+1)/u_n, where {a_n}_n is a cycle with positive values a, b and with positive initial conditions. It is known that for a = b = 1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a, b) ≠ (1, 1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a ≠ b, then any odd period, except 1, appears.

In this partly expository paper, we study the set of groups of orientation-preserving homeomorphisms of the circle S¹ which do not admit non-abelian free subgroups. We use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in . Of the known results, we include some results from a family of results of Beklaryan and Malyutin, and we also give a new proof of a theorem of Margulis. Our primary new results include a detailed classification of the solvable subgroups of R. Thompson's group T.

It is a well known result that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module; we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the single crossing tangles (⤲) and (⤲), and prove that it is preserved by appropriately "alternating" planar algebra compositions. Hence, this condition is satisfied by the Jones polynomial of all alternating tangles. Finally, in the case of 0-tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial.

We describe a "concentration on the diagonal" condition on the Khovanov complex of tangles, show that this condition is satisfied by the Khovanov complex of the single crossing tangles and , and prove that it is preserved by alternating planar algebra compositions. Hence, this condition is satisfied by the Khovanov complex of all alternating tangles. Finally, in the case of 0-tangles, meaning links, our condition is equivalent to a well-known result [E. S. Lee, The support of the Khovanov's invariants for alternating links, preprint (2002), arXiv:math.GT/0201105v1.] which states that the Khovanov homology of a non-split alternating link is supported on two diagonals. Thus our condition is a generalization of Lee's theorem to the case of tangles.

We classify Legendrian torus knots in S¹ × S² with its standard tight contact structure up to Legendrian isotopy.

For any generic immersion of a Petersen graph into a plane, the number of crossing points between two edges of distance one is odd. The sum of the crossing numbers of all $5$-cycles is odd. The sum of the rotation numbers of all $5$-cycles is even. We show analogous results for $6$-cycles, $8$-cycles and $9$-cycles. For any Legendrian spatial embedding of a Petersen graph, there exists a $5$-cycle that is not an unknot with maximal Thurston–Bennequin number, and the sum of all Thurston–Bennequin numbers of the cycles is seven times the sum of all Thurston–Bennequin numbers of the $5$-cycles. We show analogous results for a Heawood graph. We also show some other results for some graphs. We characterize abstract graphs that have a generic immersion into a plane whose all cycles have rotation number $0$.

The purpose of this paper is to introduce an infinite sequence {St_k}_k of mutually independent topological invariants of smooth closed plane curves, which is proved to be a natural extension of the rotation number and the strangeness invariant defined by Arnold in [Ar]. We prove a formula to express St_k by using the invariants which are defined in [Oz]. The jumps of St_k at perestroikas of three types (namely cusp point, triple point, and self tangent point perestroika) are investigated, and as a consequence we find that St_k have the order in the sense of Vassiliev equal to 1.

It is shown that Legendrian (respectively transverse) cable links in S³ with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J¹(S¹) with its standard tight contact structure.

In this paper, we will introduce the rotation number for the one-dimensional asymmetric p-Laplacian with a pair of periodic potentials. Two applications of this notion will be given. One is a clear characterization of two unbounded sequences of Fučik curves of the periodic Fučik spectrum of the p-Laplacian with potentials. With the help of the Poincaré–Birkhoff fixed point theorem, the other application is some existence result of multiple periodic solutions of nonlinear ordinary differential equations concerning with the p-Laplacian.

Traynor ([11]) has described an example of a two-component Legendrian "circular helix link" Λ₀ ⊔ Λ₁ in the 1-jet space J¹(S¹) of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to the link Λ₁ ⊔ Λ₀. We give a complete classification of the Legendrian realizations of this topological link type, as well as all other "cable links" in J¹(S¹).

We consider the rotation of tangent vectors and the mutual rotation of two particles undergoing motion in a random two-dimensional velocity field. We look at random fields whose laws are invariant under translations and rotations, but not reflections. These polarized fields are intermediate between homogeneous and isotropic fields, and may possess a preferred sense of rotation. The covariance of such a field must have a certain form which we describe. In a Brownian flow based on a polarized random field, we show that tangent vectors can rotate at a constant asymptotic rate, and that under certain conditions, two particles will rotate about each other at the same asymptotic rate. For illustration we present simulations of a polarized Gaussian field and of particles moving in a polarized Brownian flow.

This paper is devoted to the study of the asymptotic dynamics of a class of coupled second order oscillators driven by white noises. It is shown that any system of such coupled oscillators with positive damping and coupling coefficients possesses a global random attractor. Moreover, when the damping and the coupling coefficients are sufficiently large, the global random attractor is a one-dimensional random horizontal curve regardless of the strength of the noises, and the system has a rotation number, which implies that the oscillators in the system tend to oscillate with the same frequency eventually and therefore the so-called frequency locking is successful. The results obtained in this paper generalize many existing results on the asymptotic dynamics for a single second order noisy oscillator to systems of coupled second order noisy oscillators. They show that coupled damped second order oscillators with large damping have similar asymptotic dynamics as the limiting coupled first order oscillators as the damping goes to infinite and also that coupled damped second order oscillators have similar asymptotic dynamics as their proper space continuous counterparts, which are of great practical importance.