Chaos, cycles, and schwarzian derivatives for families of asymmetric maps

Abstract

We define dynamical systems of families of asymmetric maps on the unit interval. The asymmetric maps are derived by conjugation with Bernoulli Shifts, tent maps, logistic maps, and logistic shift maps. These maps are asymmetric Bernoulli Shifts, asymmetric tent maps, asymmetric logistic maps, and asymmetric logistic shift maps. We prove that these dynamical systems are chaotic and we compute invariant densities and Lyapunov exponents. We investigate properties of the Schwarzian derivatives in these families of functions.

Keywords:

• discrete dynamical systems
• Schwarzian derivatives
• asymmetric maps