Sensitive Dependence and Dense Periodic Points

We show sensitive dependence on initial conditions and dense set of periodic points imply asymptotic sensitivity, a stronger form of sensitivity , where the deviation happens not just once but infinitely many times. As a consequence it follows that all Devaney chaotic systems (e.g. logistic map) have this asymptotic sensitivity. Sensitive dependence on initial conditions (shortly, sensitivity) is a central concept in the theory chaos and discrete dynamical systems. Roughly speaking, a sensitive system with a sensitivity constant δ > 0 has the following property. Arbitrarily close to the initial point of any specified trajectory, there is a point whose trajectoty deviates from that specified one by a distance more than δ, atleast at one instant of time later. The words arbitrarily close emphasizes the fact that even by choosing the initial point closer and closer to that specified inital point one cannot avoid a deviation in distance more than δ. Thus, the arbitrarily small initial separation grows up to more than δ > 0, within a finite time. Hence, in predicting the future an error of magnitude δ is inevitable, however small be the error in the initial condition. This is often refered to, rather crudely, as the divergence of nearby trajecto-ries. The following definition of sensitivity makes this precise. Discrete dynamical system is a pair (X, T) where X is a metric space and T is a continuous self-map on X. B r (x) denotes the open ball centered at x and of radius r. The distance between the points x and y of the space X is denoted by d(x, y). The space X is supposed to represent all possible states of a physical system and T models the evolution of the system from the present state x to the next state T (x). Thus the sequence {x, T (x), T 2 (x), T 3 (x), ...} represents the evolution of the state x and is called the trajectory of x. If T k (x) = x for some natural number k then x is periodic point. The smallest k for which T k (x) = x is the order of the periodic point x.