On the existence of a new type of periodic and quasi-periodic orbits for twist maps of the torus

We prove that for a large and important class of $C^1$ twist maps of the torus periodic and quasi-periodic orbits of a new type exist, provided that there are no rotational invariant circles (R.I.C's). These orbits have a non-zero ''vertical rotation number'' (V.R.N.), in contrast to what happens to Birkhoff periodic orbits and Aubry-Mather sets. The V.R.N. is rational for a periodic orbit and irrational for a quasi-periodic. We also prove that the existence of an orbit with a $V.R.N=a>0,$ implies the existence of orbits with $V.R.N=b,$ for all $0<b<a.$ In this way, related to a generalized definition of rotation number, we characterize all kinds of periodic and quasi-periodic orbits a twist map of the torus can have. And as a consequence of the previous results we obtain that a twist map of the torus with no R.I.C's has positive topological entropy, which is a very classical result. In the end of the paper we present some examples, like the Standard map, such that our results apply.Comment: 20 pages. to appear in Nonlinearity 15(5) 1399-141

Don’t curse the inflow of emails: It can help you do your job better

When email interruptions are congruent with our core responsibilities, they help us process tasks mindfully, writes Shamel Adda

Persistence of fixed points under rigid perturbations of maps

Let $f:S^1\times [0,1]\to S^1\times [0,1]$ be a real-analytic annulus diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift $\tilde {f}:\mathbb{R}\times [0,1]\rightarrow \mathbb{R}\times [0,1]$ we have ${\rm Fix}(\tilde{f})=\mathbb{R}\times \{0\}$ and that $\tilde{f}$ positively translates points in $\mathbb{R}\times \{1\}$. Let $\tilde{f}_\epsilon$ be the perturbation of $\tilde{f}$ by the rigid horizontal translation $(x,y)\mapsto (x+\epsilon,y)$. We show that for all $\epsilon >0$ sufficiently small we have ${\rm Fix} (\tilde{f}_\epsilon)=\emptyset$. The proof follows from Ker\'ekj\'art\'o's construction of Brouwer lines for orientation preserving homeomorphisms of the plane with no fixed points. This result turns out to be sharp with respect to the regularity assumption: there exists a diffeomorphism $f$ satisfying all the properties above, except that $f$ is not real-analytic but only smooth, so that the above conclusion is false. Such a map is constructed via generating functions

Dynamics of homeomorphisms of the torus homotopic to Dehn twists

In this paper we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential "horizontal" set K, invariant under $f$. In other words, if we consider the lift $\hat{f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat{f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland's conjecture to this setting: If $f$ is area preserving and has a lift $\hat{f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat{f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity

A Call for Engaging Context in HCI/MIS Research with Examples from the Area of Technology Interruptions

This paper contributes to the discussion on future directions of Human-Computer Interaction in Information Systems (HCI/MIS) research by explicating the role of task- and social context. We show that context has not been sufficiently engaged, and argue why it is important to pay more attention to it in theory and design of future HCI/MIS research. Drawing on examples from the core HCI area of technology interruptions, we formulate a set of general research questions and guidelines, which allow us to represent the context of multiple users in continuous collaboration with multiple tools while working on tasks that are intertwined within business processes. These guidelines will generate new insights for HCI/MIS research and allow us to develop research that captures the changing nature of the computing environment

Landscape Architecture and The Saudi Arabia Quality of Life Program

Saudi Arabia has for decades felt the effects of the declining condition of its urban, social and natural environments. The government has set out a long-term vision to address these issues through 12 major programs, one of which is the Quality of Life Program 2020. The Program mainly focuses on making Saudi Arabia a top living destination by improving individuals’ lifestyles and enhancing their quality of life. This paper considers the importance of landscape architecture to the Program by way of a literature review to clarify the role of the landscape architecture profession and an analysis of the projects that underlie the Program to highlight the profession’s potential to contribute to these projects