Classical dynamical systems

[Daniel]

Stephen Wolfram was interested in extending tetration because of what it might tell us about tell about fractional iteration. There are two solid mathematical systems for doing physics, PDE's and iterated functions. The problem is that time in physics appears to be continuous while in iterated functions it appears discrete. Thus is the importance of the work we do here in the context of physics.

Quoting Arnold and Avez, 1968 in E. Lee Jackson's Perspectives of nonlinear dynamics, Vol 1., pg. 51

Let M be a smooth manifold, \( \mu \) a measure on M defined by a continuous positive density,
\(f^t:M \to M\)
a one-parameter group of measure-preserving diffeomorphisms. The collection
\((M,\mu, f^t)\) is called a classical dynamical system.