ON THE SIGNIFICANCE OF THE TENT MAP

Abstract

Discrete time dynamical systems generated by the iteration of nonlinear maps provide simple and interesting examples of chaotic systems. But what is the physical principle behind the emergence of these maps? In this note we present an approach to this problem by considering a class, Y, of 2–1 chaotic maps on [0, 1] that are symmetric and have symmetric invariant densities. We prove that such maps are conjugate to the tent map. In Y we search for maps that minimizes a functional that depends on y ∈ Y and fy, the probability density function invariant under y. We define a simple functional whose extremal value is achieved by the tent map.