On the omega-limit sets of tent maps

For a continuous map f on a compact metric space (X,d), a subset D of X is internally chain transitive if for every x and y in D and every delta > 0 there is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) < delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed subset D of X is internally chain transitive if and only if D is an omega-limit set for some point in X, and that the same is also true for the tent map with slope equal to 2. In this paper, we prove that for tent maps whose critical point c=1/2 is periodic, every closed, internally chain transitive set is necessarily an omega-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an omega-limit set. Together, these results lead us to conjecture that for those tent maps with shadowing (or pseudo-orbit tracing), the omega-limit sets are precisely those sets having internal chain transitivity.Comment: 17 page

Shadowing, asymptotic shadowing and s-limit shadowing

We study three notions of shadowing: classical shadowing, limit (or asymptotic) shadowing, and s-limit shadowing. We show that classical and s-limit shadowing coincide for tent maps and, more generally, for piecewise linear interval maps with constant slopes, and are further equivalent to the linking property introduced by Chen in 1991. We also construct a system which exhibits shadowing but not limit shadowing, and we study how shadowing properties transfer to maximal transitive subsystems and inverse limits (sometimes called natural extensions). Where practicable, we show that our results are best possible by means of examples.Comment: 28 pages, 4 figure

Design and implementation of sensor systems for control of a closed-loop life support system

The sensing and controlling needs for a Closed-Loop Life Support System (CLLSS) were investigated. The sensing needs were identified in five particular areas and the requirements were defined for workable sensors. The specific areas of interest were atmosphere and temperature, nutrient delivery, plant health, plant propagation and support, and solids processing. The investigation of atmosphere and temperature control focused on the temperature distribution within the growth chamber as well as the possibility for sensing other parameters such as gas concentration, pressure, and humidity. The sensing needs were studied for monitoring the solution level in a porous membrane material along with the requirements for measuring the mass flow rate in the delivery system. The causes and symptoms of plant disease were examined and the various techniques for sensing these health indicators were explored. The study of sensing needs for plant propagation and support focused on monitoring seed viability and measuring seed moisture content as well as defining the requirements for drying and storing the seeds. The areas of harvesting, food processing, and resource recycling, were covered with a main focus on the sensing possibilities for regulating the recycling process