One-Dimensional Dynamical Systems

Ana Rodrigues is an associate professor in the Mathematics Department, University of Exeter. She earned her PhD in mathematics in dynamical systems in 2007 from the University of Porto. Before arriving at Exeter, she was a postdoc at Indiana University Purdue University at Indianapolis, USA, for two years and then held a research assistant position at KTH - Royal Institute of Technology and Uppsala University, Sweden, financed by the Swedish Research Council. Her research interests are in dynamical systems (low-dimensional dynamical systems, ergodic theory, limit cycles of differential equations and dynamical systems with symmetry).

1. Introduction. 2. Rotation Numbers. 2.1. Arnold Tongues for Double Standard Maps. 2.2 Arnold Tongues for ɑ-Standard Maps. 3. Topological conjugacy. 4. Critical points. 5. Topological theory of Chaos. 5.1. Topological Entropy. 5.2. Schwarzian Derivative. 6. Symbolic Dynamics. 6.1. Kneading Sequences for Double Standard Maps. 6.2 Kneading Sequences for ɑ-Standard Maps. 7. Tongues. 7.1. Length of Tongues. 7.2. Boundary of The Tongues. 7.3. Tip of the Tongues. 7.4. Connectedness of Tongues. 7.5. Arnold Tongues of Higher Periods for ɑ-Standard Maps. Bibliography.