1 Dynamics on the Irrationals

We present here work which is, in part, expository with proofs, exercises (2.4, 2.6, 4.3, and 7.6), and, in part, contains new results (3.1, 5.4, amd 7.7) so we ought to begin with some background: Dynamics travels a line of history from as far back as Newton, as a notion for his laws of motion, especially, as concerns the law of gravity, for which he developed the Calculus. Prior to the twentieth century, a dynamical system meant a motion whose parameters are functions of time and satsfy a system of differential equations. Eighteenth and Nineteenth century analysts used various analytical manipulations (including infinte series) to cause the differential equations to reveal information. However, 100 years ago Poincaré, using a proofs fortelling modern topology, shifted our attention from particular solutions to the relationships between all possible solutions and, in some cases, he used his methods to prove the existence of periodic solutions. In 1927, G.D. Birkhoff's work signifcantly justified Poincare's global approach by proving that in any dynamical system on a compact space has a solution stable in the sense of Poisson. In our terminolgy it is stated as each system on a compact space has a recurrent point.