Time-one Map Research Articles

Non-integrability vs. integrability in pentagram maps

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability.

Farey maps, Diophantine approximation and Bruhat–Tits tree

Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents via the symbolic coding of the geodesic flow of the continued fraction algorithm for formal power series with coefficients in a finite field, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-one map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.

Topological contact dynamics II: Topological automorphisms, contact homeomorphisms, and non-smooth contact dynamical systems

This sequel to our previous paper [MS11b] continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems. The article includes an examination of the groups of time-one maps of topological contact and strictly contact isotopies, and the construction of a bi-invariant metric on the latter. Moreover, every topological contact or strictly contact dynamical system is arbitrarily close to a continuous contact or strictly contact dynamical system with the same end point. In particular, the above groups of time-one maps are independent of the choice of norm in the definition of the contact distance. On every contact manifold we construct topological contact dynamical systems with time-one maps that fail to be Lipschitz continuous, and smooth contact vector fields whose flows are topologically conjugate but not conjugate by a contact C^1-diffeomorphism.

Embedding in a flow of Morse-Smale diffeomorphisms on manifolds of dimension higher than two

We say that a diffeomorphism f : Mn → Mn embeds in a topological (smooth) flow if there exists a topological (smooth) flow Xt on Mn such that f is the time-one map of Xt. It follows from results of the papers [1] and [2], in which the structural stability of Morse–Smale diffeomorphisms was proved, that, for any manifold Mn, there exists an open (in Diff(Mn)) set of Morse–Smale diffeomorphisms embeddable in a topological flow. In [1], Palis proved the following necessary condition for a Morse– Smale diffeomorphism f to embed in a topological flow:

Weak mixing suspension flows over shifts of finite type are universal

Let $S$ be an ergodic measure-preserving automorphism on a nonatomicprobability space, and let $T$ be the time-one map of a topologically weakmixing suspension flow over an irreducible subshift of finite type under aHölder ceiling function. We show that if the measure-theoretic entropyof $S$ is strictly less than the topological entropy of $T$, then thereexists an embedding of the measure-preserving automorphism into thesuspension flow. As a corollary of this result and the symbolic dynamicsfor geodesic flows on compact surfaces of negative curvature developed byBowen [5] and Ratner [31], we also obtain anembedding of the measure-preserving automorphism into a geodesic flowwhenever the measure-theoretic entropy of $S$ is strictly less than thetopological entropy of the time-one map of the geodesic flow.

Connecting orbits for families of Tonelli Hamiltonians

We investigate the existence of Arnold diffusion-type orbits forsystems obtained by iterating in any order the time-one maps of afamily of Tonelli Hamiltonians. Such systems are known as'polysystems' or 'iterated function systems'. When specialized tofamilies of twist maps on the cylinder, our results are similar tothose obtained by Moeckel [20] and Le Calvez [15]. Ourapproach is based on weak KAM theory and is close to the one used byBernard in [3] to study the case of a single TonelliHamiltonian.

Transitive Anosov flows and Axiom-A diffeomorphisms

AbstractLet M be a smooth compact Riemannian manifold without boundary, and ϕ:M×ℝ→M a transitive Anosov flow. We prove that if the time-one map of ϕ is C1-approximated by Axiom-A diffeomorphisms with more than one attractor, then ϕ is topologically equivalent to the suspension of an Anosov diffeomorphism.

COMPLEXITY OF CONTINUOUS SEMI-FLOWS AND RELATED DYNAMICAL PROPERTIES

The equicontinuity and scattering properties of continuous semi-flows are studied on a compact metric space. The main results are obtained as follows: first, the complexity function defined by the spanning set is bounded if and only if the system is equicontinuous; secondly, if a continuous semi-flow is topologically weak mixing, then it is pointwise scattering; thirdly, several equivalent conditions for the time-one map of a continuous semi-flow to be scattering are presented; Finally, for a minimal continuous map it is shown that the "non-dense" requirement is unnecessary in the definition of scattering by using open covers.

Analytic invariants associated with a parabolic fixed point in 2

It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (ℝ2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in ℂ2 is not available at the present time.

Holomorphic germs and the problem of smooth conjugacy in a punctured neighborhood of the origin

We consider germs of conformal mappings tangent to the identity at the origin in C. We construct a germ of a homeomorphism which is a C ∞ diffeomorphism except at the origin conjugating these holomorphic germs with the time-one map of the vector field V(z) = z m ∂ ∂z. We then show that, in the case m = 2, for a germ of a homeomorphism which is real-analytic in a punctured neighborhood of the origin, with real-analytic inverse, conjugating these germs with the time-one map of the vector field exists if and only if a germ of a biholomorphism exists.

Invertible Contractions and Asymptotically Stable ODE’S that are not $$\mathcal{C}^{1}$$ -Linearizable

We present an example of a contraction diffeomorphism in infinite dimensions that is not $$\mathcal {C}^{1}$$ -linearizable, and we construct a regular ordinary differential equation in a Hilbert space whose time-one map is that diffeomorphism. With this we have an example of an asymptotically stable ODE that is not $$\mathcal {C}^{1}$$ -conjugate to its linear part.

Floer--Novikov cohomology and symplectic fixed points

We introduce variants of Floer-Novikov cohomology and give a lower bound for the number of fixed points of time-one maps of time-dependent symplectic flows under the non-degeneracy condition.

Neimark–Sacker bifurcation for periodic delay differential equations

In this paper we study the delay differential equation x ˙ ( t ) = γ ( a ( t ) x ( t ) + f ( t , x ( t - 1 ) ) ) , where γ is a real parameter, the functions a ( t ) , f ( t , ξ ) are C 4 -smooth and periodic in the variable t with period 1. Varying the parameter, eigenvalues of the monodromy operator (the derivative of the time-one map at the equilibrium 0) cross the unit circle and bifurcation of an invariant curve occurs. To detect the critical parameter-values, we use Floquet theory. We give an explicit formula to compute the coefficient that determines the direction of the bifurcation. We extend the center manifold projection method to our infinite-dimensional Banach space using spectral projection represented by a Riesz–Dunford integral. The Neimark–Sacker Bifurcation Theorem implies the appearance of an invariant torus in the space C × S 1 . We apply our results to an equation used in neural network theory.

Statistical properties of compact group extensions of hyperbolic flows and their time one maps

Recent work of Dolgopyat shows that'typical' hyperbolic flows exhibit rapid decay of correlations.Melbourne and Török used this result to derive statisticallimit laws such as the central limit theorem and the almost sureinvariance principle for the time-one map of such flows.In this paper, we extend these results to equivariant observationson compact group extensions of hyperbolic flows and their time onemaps.

Stable Ergodicity and Frame Flows

In this note we show that all diffeomorphisms close enough to the time-one map of the frame flow on certain negatively curved manifolds are ergodic. As a simple corollary we deduce that the frame flows are ergodic for all compact manifolds with curvature pinched sufficiently close to −1, thus providing results in the case of manifolds of dimension 7 or 8 which were missing from the results of Brin and Karcher.

On spreading speeds and traveling waves for growth and migration models in a periodic habitat.

It is shown that the methods previously used by the author [Wei82] and by R. Lui [Lui89] to obtain asymptotic spreading results and sometimes the existence of traveling waves for a discrete-time recursion with a translation invariant order preserving operator can be extended to a recursion with a periodic order preserving operator. The operator can be taken to be the time-one map of a continuous time reaction-diffusion model, or it can be a more general model of time evolution in population genetics or population ecology in a periodic habitat. Methods of estimating the speeds of spreading in various directions will also be presented.

A note on diffeomorphism and periodic differential equations

It is shown that diffeomorphisms are equivalent to time-one maps of one-time periodic differential equations. As an application, we show how this is related to numerical schemes.

Central Limit Theorems and Invariance Principles¶for Time-One Maps of Hyperbolic Flows

We give a general method for deducing statistical limit laws in situations where rapid decay of correlations has been established. As an application of this method, we obtain new results for time-one maps of hyperbolic flows.

Entropy of flows, revisited

We introduce a concept of measure-theoretic entropy for flows and study its invariance under measure-theoretic equivalences. Invariance properties of the corresponding topological entropy is studied too. We also answer a question posed by Bowen-Walters in [3] concerning the equality between the topological entropy of the time-one map of an expansive flow and the time-one map of its symbolic suspension.

Hamiltonian suspensions of symplectomorphisms: an alternative approach to design problems

We solve the problem that often arises in optics and charged particle transport design tasks of finding a time-dependent Hamiltonian system that produces a given symplectic map. In addition, we show how, given a particular Hamiltonian, to produce many Hamiltonians that have the same time-one map so that one has a large class of Hamiltonians from which to choose the ‘best’ design. We also show how to produce many symplectic maps that are given explicitly rather than implicitly so that the design problem is more tractable.