A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.Comment: Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue 1-2, pp 405-42

A toral diffeomorphism with a non-polygonal rotation set

We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon

Expansive homeomorphisms of the plane

This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques involve topological and metric aspects of the plane. The use of a Lyapunov metric function which defines the same topology as the one induced by the usual metric but that, in general, is not equivalent to it is an example of such techniques. The discovery of a hypothesis about the behavior of Lyapunov functions at infinity allows us to generalize some results that are valid in the compact context. Additional local properties allow us to obtain another classification theorem.Comment: 29 pages, 22 figure

Topological Entropy of Braids on the Torus

A fast method is presented for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle trajectories. Our approach is a generalization of Moussafir's technique for braids on the sphere. Previous methods for computing topological entropies include the Bestvina--Handel train-track algorithm and matrix representations of the braid group. However, the Bestvina--Handel algorithm quickly becomes computationally intractable for large braid words, and matrix methods give only lower bounds, which are often poor for large braids. Our method is computationally fast and appears to give exponential convergence towards the exact entropy. As an illustration we apply our approach to the braiding of both periodic and aperiodic trajectories in the sine flow. The efficiency of the method allows us to explore how much extra information about flow entropy is encoded in the braid as the number of trajectories becomes large.Comment: 19 pages, 44 figures. SIAM journal styl

Strictly Toral Dynamics

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus $\mathbb{T}^2$ which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points $ine(f)$ is shown to be a disjoint union of periodic topological disks ("elliptic islands"), while the set of essential points $ess(f)$ is an essential continuum, with typically rich dynamics (the "chaotic region"). This generalizes and improves a similar description by J\"ager. The key result is boundedness of these "elliptic islands", which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in $ess(f)$ is as rich as in $\mathbb{T}^2$ from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.Comment: Incorporates suggestions and corrections by the referees. To appear in Inv. Mat

Comparison of Summer Forages and the Effect of Nitrogen Fertilizers on \u3ci\u3eBrassica\u3c/i\u3e Forages in Tasmania

Summer forage crops, and in particular Brassica spp., have become increasingly popular in dairy production systems in Tasmania. Field experiments were conducted for 3 years in northwestern Tasmania, in the spring/summers beginning in 1995. The study aimed to compare yield and quality of Brassica and Poaceae forages and the response of Brassica species to nitrogen (N) (50, 100 and 200 kg N/ha) and irrigation. The average total yields of dryland (rainfed) crops in 1995 to 1997 experiments, were turnip (Brassica rapa) 9.3 t/ha, rape (B. napus) 5.9 t/ha, oats (Aevena sativa) 5.2 t/ha, kale (B. oleracea) 5.1 t/ha, short-lived ryegrass (Lolium multiflorum) 5.1 t/ha, pasja (B. campestris × B. napus) 4.3 t/ha, perennial ryegrass (L. perenne) 4.2 t/ha, millet (Echinochloa utilis) 3.8 t/ha, and maize. (Zea mays) 2.9 t/ha. Irrigation increased the yield of turnips by 4.8 t/ha (mainly bulbs) and millet yields by 1.4 t/ha and reduced maize yield by 1.2 t/ha. Brassica species were higher in ME and lower in CP than the Poaceae forages. Nitrogen fertilizer increased the DM yield of tops of all Brassica crops in the 1997/98 experiments under irrigation, but it decreased the yield of turnips bulbs. The total yields with 50, 100 and 200 kg N/ha were 14, 15.2 and 15 t DM/ha for turnips, 7.5, 8.5 and 10 t for pasja and 10, 12 and 12.2 t DM/ha for rape, respectively. With 100 kg N/ha the average concentration of quality attributes of turnips, pasja and rape were CP 14, 22 and 19%, ME 12, 14.6 and 12.6 MJ/kg DM respectively. Nitrogen increased the CP, but had no effect on ME of any Brassica crops. Brassica forage are superior to Poaceae forages for summer feed production and as a part of pasture renovation process. They are higher in their yields, quality and water use efficiency and respond well to N fertilizer

Aperiodic invariant continua for surface homeomorphisms

We prove that if a homeomorphism of a closed orientable surface S has no wandering points and leaves invariant a compact, connected set K which contains no periodic points, then either K=S and S is a torus, or $K$ is the intersection of a decreasing sequence of annuli. A version for non-orientable surfaces is given.Comment: 8 pages, to appear in Mathematische Zeitschrif

Periodic orbits of period 3 in the disc

Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.Comment: 7 page

Necessity of integral formalism

To describe the physical reality, there are two ways of constructing the dynamical equation of field, differential formalism and integral formalism. The importance of this fact is firstly emphasized by Yang in case of gauge field [Phys. Rev. Lett. 33 (1974) 445], where the fact has given rise to a deeper understanding for Aharonov-Bohm phase and magnetic monopole [Phys. Rev. D. 12 (1975) 3845]. In this paper we shall point out that such a fact also holds in general wave function of matter, it may give rise to a deeper understanding for Berry phase. Most importantly, we shall prove a point that, for general wave function of matter, in the adiabatic limit, there is an intrinsic difference between its integral formalism and differential formalism. It is neglect of this difference that leads to an inconsistency of quantum adiabatic theorem pointed out by Marzlin and Sanders [Phys. Rev. Lett. 93 (2004) 160408]. It has been widely accepted that there is no physical difference of using differential operator or integral operator to construct the dynamical equation of field. Nevertheless, our study shows that the Schrodinger differential equation (i.e., differential formalism for wave function) shall lead to vanishing Berry phase and that the Schrodinger integral equation (i.e., integral formalism for wave function), in the adiabatic limit, can satisfactorily give the Berry phase. Therefore, we reach a conclusion: There are two ways of describing physical reality, differential formalism and integral formalism; but the integral formalism is a unique way of complete description.Comment: 13Page; Schrodinger differential equation shall lead to vanishing Berry phas

Dinoflagellate blooms and physical systems in the Gulf of Maine

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution May 1990Numerous studies have shown dinoflagellate blooms to be closely related to density discontinuities and fronts in the ocean. The spatial and temporal patterns of the dinoflagellate population depend on the predominant mode of physical forcing, and its scales of variability. The present study combined field sampling of hydrographic and biological variables to examine the relationship of dinoflagellate population distributions to physical factors along the southwestern cost of the Gulf of Maine. A bloom of Ceratium longipes occurred along this coast during the month of June, 1987. A simple model which coupled along-isopycnal diffusion with the logistic growth equation suggested that the cells had a growth rate of about 0.1 d-1 , and had reached a steady horizontal across-shelf distribution within about 10 d. Fur~her variations in population density appeared to be related to fluctuations of light with periods of -10 d. To our knowledge, this was the first use of this simple diffusion model as a diagnostic tool for quantifying parameters describing the growth and movement of a specific phytoplankton population. Blooms of the toxic dinoflagellate, Alexandrium tamarense have been nearly annual features along the coasts of southern Maine, New Hampshire and Massachusetts since 1972; however the mechanisms controlling the distribution of cells and concomitant shellfish toxicity are relatively poorly understood. Analysis of field data gathered from April to September, 1987-1989, showed that in two years when toxicity was detected in the southern part of this region, A. tamarense cells were apparently transported into the study area between Portsmouth and Cape Ann, Massachusetts, in a coastally trapped buoyant plume. This plume appears to have been formed off Maine by the outflow from the Androscoggin and Kennebec Rivers. Flow rates of these rivers, hydrographic sections, and satellite images suggest that the plume had a duration of about a month, and extended alongshore for several hundred kilometers. The distribution of cells followed the position of the plume as it was influenced by wind and topography. Thus when winds were downwelling-favourable, cells were moved alongshore to the south, and were held to the coast; when winds were upwelling-favourable, the plume sometimes separated from the coast, advecting the cells offshore. The alongshore advection of toxic cells within a coastally trapped buoyant plume can explain the temporal and spatial patterns of shellfish toxicity along the coast. The general observation of a north-to-south temporal trend of toxicity is consistent with the southward advection of the plume. In 1987 when no plume was present, Alexandrium tamarense cells were scarce, and no toxicity was recorded at the southern stations. A hypothesis was formulated explaining the development and spread of toxic dinoflagellate blooms in this region. This plume-advection hypothesis included: source A. tamarense populations in the north, possibly associated with the Androscoggin and Kennebec estuaries; a relationship between toxicity patterns and river flow volume and timing of flow peaks; and a relationship between wind stresses and the distribution of low salinity water and cells. Predictions of the plume-advection hypothesis were tested with historical records of shellfish toxicity, wind speed and direction, and river flow. The predictions tested included the north-south progression of toxic outbreaks, the occurrence of a peak in river flow prior to the PSP events, the relationship of transit time of PSP toxicity along the coast with river flow volume, and the influence of surface wind stress on the timing and location of shellfish toxicity. All the predictions tested were supported by the historical records. In addition it was found that the plume-advection hypothesis explains many details of the timing and spread of shellfish toxicity, including the sporadic nature of toxic outbreaks south of Massachusetts Bay, and the apparently rare occurrence of toxicity well offshore on Nantucket Shoals and Georges Bank.This research was supported by ONR contract N00014-87-K-0007 and ONR grant N00014-89-J-111 to Donald M. Anderson, and NOAA Office of Sea Grant contract NA86AA-D-SG090