Dynamical system theory invariant measures

Chapter 4 covers much of the same ground as chapter 3 but from a more formal dynamical systems theory approach. The discrete CA world is examined in the context of what is known about the behavior of continuous dynamical systems, and a number of important methodological tools developed by dynamical systems theory (i.e. Lyapunov exponents, invariant measures, and various measures of entropy and... 

For the reader unfamiliar with Liapunov theory, the intuition goes like this V is some sort of measure of height. The magnitude of dV/dt = VV-/< 0 is a measure of how fast solutions run downhill. The downhill slide stops at E. The theory of dynamical systems says that the omega limit set is invariant and therefore contained in M. 

For the most satisfactory and coherent system of dynamics and physics the conservation of momentum must be retained, and what now emerges is that even in the modified dynamical theory (special theory of relativity) the observers A and B would both verify the constancy of momentum if the mass of a body were itself assumed no longer to be invariable but to be a function of its velocity in the particular system where the measurement is made. Einstein concluded that there would be a variation according to the equation... 

Differential equations of pure mechanical systems generate transformation groups for which the Lebesgue measure is invariant this statement is called the Liouville theorem. Major results of the modern theory of dynamic systems are connected with physical sciences, mostly with mechanics. Differential equations of physics may refer to particle or planetary motions described by ordinary differential equations, or to wave motion described by partial differential equations. Dissipative effects are neglected in all these systems, and so the emphasis is on conservative or Hamiltonian systems. 

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