Cylindrical contact homology and topological entropy

The main purpose of this paper is to investigate a parametric topological entropy for impulsive differential inclusions on tori. In this way, besides other matters, we would like to extend our recent results concerning impulsive differential equations as well as those on “nonparametric” topological entropy to impulsive differential inclusions. Parametric topological entropy, which is usually called a topological entropy for nonautonomous dynamical systems, is considered here via the compositions of associated multivalued Poincaré translation operators with the single-valued time-dependent impulsive maps. On compact polyhedra and, in particular on tori, parametric topological entropy for families of admissible multivalued maps can be estimated from below by means Ivanov-type inequality in terms of the asymptotic Nielsen and Lefschetz numbers which are, unlike the topological entropy, homotopy invariants. In the scalar case, an effective criterion for a positive parametric topological entropy can be given by topological degree arguments for equi-continuous impulsive maps. In a single-valued nonparametric case, a positive topological entropy usually signifies topological chaos. Some simple illustrative examples are provided.

This paper is the sequel to the previous paper [Nelson, Abh. Math. Semin. Univ. Hambg. 85 (2015) 125–179], which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically convex contact forms. The Reeb orbits of these so-called dynamically separated contact forms satisfy a uniform growth condition on their Conley–Zehnder indices with respect to a free homotopy class. Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show how to obtain the desired cylindrical contact homology by taking direct limits. We give a direct proof of invariance of cylindrical contact homology within the class of dynamically separated contact forms, and elucidate the independence of the filtered cylindrical contact homology with respect to the choice of the dynamically separated contact form and almost complex structure. We also show that these regularity results are compatible with geometric methods of computing cylindrical contact homology of prequantization bundles, proving a conjecture of Eliashberg [Symplectic field theory and its applications, International Congress of Mathematicians I (European Mathematical Society, Zürich, 2007) 217–246] in dimension three.

Topological entropy of a dynamical system is an upper bound for the sum of positive Lyapunov exponents; in practice, it is strongly indicative of the presence of mixing in a subset of the domain. Topological entropy can be computed by partition methods, by estimating the maximal growth rate of material lines or other material elements, or by counting the unstable periodic orbits of the flow. All these methods require detailed knowledge of the velocity field that is not always available, for example, when ocean flows are measured using a small number of floating sensors. We propose an alternative calculation, applicable to two-dimensional flows, that uses only a sparse set of flow trajectories as its input. To represent the sparse set of trajectories, we use braids, algebraic objects that record how trajectories exchange positions with respect to a projection axis. Material curves advected by the flow are represented as simplified loop coordinates. The exponential rate at which a braid stretches loops over a finite time interval is the Finite-Time Braiding Exponent (FTBE). We study FTBEs through numerical simulations of the Aref Blinking Vortex flow, as a representative of a general class of flows having a single invariant component with positive topological entropy. The FTBEs approach the value of the topological entropy from below as the length and number of trajectories is increased; we conjecture that this result holds for a general class of ergodic, mixing systems. Furthermore, FTBEs are computed robustly with respect to the numerical time step, details of braid representation, and choice of initial conditions. We find that, in the class of systems we describe, trajectories can be re-used to form different braids, which greatly reduces the amount of data needed to assess the complexity of the flow.

We consider a [Formula: see text] free action [Formula: see text] on an [Formula: see text]-dimensional [Formula: see text] compact connected manifold [Formula: see text]. The manifold [Formula: see text] is foliated by circles if all orbits induced by [Formula: see text] are closed. We handle the periods of circle foliations, especially when the circles are closed geodesics with respect to some Riemannian metric. Then, it is known that the periods of [Formula: see text] are exactly the same if and only if there exists a Riemannian metric on [Formula: see text] with respect to which the circles are embedded as totally geodesic submanifolds of [Formula: see text] [A. W. Wadsley, Geodesic foliations by circles, J. Differ. Geom. 10 (1975) 541–549.]. It automatically follows a classical result [A. Besse, Manifolds All of Whose Geodesics are Closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 93 (Springer-Verlag, New York, 1978)] that all periods are exactly the same if all the geodesic flows on the cosphere bundle [Formula: see text] are periodic (the so-called “[Formula: see text]-manifold”). Our purpose here is to adapt these results to Reeb or Hamiltonian flows on contact manifolds. Then [Formula: see text]-circles are considered as totally geodesic submanifolds. Consequently, all periods are exactly the same if the contact manifold is connected and all orbits on the contact manifold are closed. We also present concrete examples of periodic flows, all of whose orbits are closed, such as Harmonic oscillators, Lotka–Volterra systems, and others. Lotka–Volterra systems, Reeb flows, and some geodesic flows have non-trivial periods, whereas the periods of Harmonic oscillators and similar systems can be easily obtained through direct calculations. As an application to quantum mechanics, we examine the spectrum of semiclassical Schrödinger operators [Formula: see text] If the spectrum of [Formula: see text] near [Formula: see text] consists of eigenvalues, we denote such a set as [Formula: see text] for small [Formula: see text]. For fixed constants [Formula: see text] and [Formula: see text], the cluster of the spectrum near [Formula: see text] (the so-called essential difference spectrum) is defined by [Formula: see text] where the notation [Formula: see text] denotes the set of cluster (accumulation) points. Under suitable assumptions, we find that [Formula: see text] In the former case, at least one of the Hamiltonian flows is not periodic on the energy surface [Formula: see text]. In the latter case, all Hamiltonian flows on [Formula: see text] are periodic with a common period [Formula: see text]. This fact represents one of the semiclassical analogies of the Helton-Guillemin theorem [V. Guillemin, Lectures on spectral theory of elliptic operators, Duke Math. J. 44 (1977) 487–517; W. Helton, An operator algebra approach to partial differential equations; Propagation of singularities and spectral theory, Indiana Univ. Math. J. 26 (1977) 997–1018.].

We construct symbolic dynamics on sets of full measure (with respect to an ergodic measure of positive entropy) for C^{1+\epsilon} flows on closed smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a closed C^\infty surface has at least const \times e^{hT}/T simple closed orbits of period less than T , whenever the topological entropy h is positive—and without further assumptions on the curvature.

We consider a semi-Riemannian metric whose associated geodesic flow either contains a non-hyperbolic periodic orbit or has infinitely many hyperbolic periodic orbits. Under some conditions, we show that the metric can be C 2 C^2 -perturbed such that the geodesic flow exhibits positive topological entropy, there are infinitely many non-lightlike closed geodesics, and their number grows exponentially with respect to the length.

We consider piecewise monotone (not necessarily, strictly) piecewise C2 maps on the interval with positive topological entropy. For such a map f we prove that its topological entropy htop(f) can be approximated (with any required accuracy) by restriction on a compact strictly f-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.

On topological entropy (chaos) realized far from fixed point

Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms

Following Knieper and Weiss [9] we exhibit explicit real analytic metrics onS2 andRP2 with positive curvature and positive topological entropy using the dynamics of the rigid body.

Positive topological entropy and Δ-weakly mixing sets

Syndetically proximal pairs

In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the ‘mixed’ sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact 3-manifolds with isometric Reeb flows (also known as R-contact manifolds) up to diffeomorphism.

In this paper, some properties of a strictly A A -coupled-expanding map in compact subsets of a metric space are studied, where A A is a transition matrix. It is shown that this map has a compact invariant set on which it is topologically semi-conjugate to the subshift for A A . If the subshift for A A has positive topological entropy, then the map is chaotic in the sense of Li-Yorke. Moreover, in the one-dimensional case, the map is at most two-to-one conjugate to the subshift for A A and chaotic in the sense of Devaney.

Complex chaotic dynamics is studies in the van der Pol oscillator model of business cycle forced by a sinusoidal function. The famous topological horseshoe theorem is applied to prove the existence of chaos from the mathematical viewpoint in this business cycle model. A rigorous proof for the existence of topological horseshoe is given. This technique combines the topological theory with the computer-assisted computations. A proper Poincare section is first chosen to obtain the corresponding Poincare map, which is proved to be semi-conjugate to the 2 -shift map. This result implies that the business cycle model has positive topological entropy, and thus is definitely chaotic.