Statistical Mechanics of Deterministic Chaos: The Case of One-Dimensional Discrete Process

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Abstract
The statistical mechanics of deterministic chaos is constructed for the case of a one-dimensional discrete process. The statistics are based on the infinite set of fixed points, which are invariants in the phase space. Consideration of the dynamical behaviour leads to the notion of the residence probability of a non-periodic orbit in the neighbourhood of a fixed point. This is turn leads to the specification of a proper probability distribution, for which the agreement with that generated by time sequence is demonstrated for several examples. The measure theoretic entropy is also calculated and is found to be equal to the Liapounov characteristic number under a condition which is satisfied in most cases. Related to this, it is demonstrated that the extended Ruelle-Bowen variational principle is valid in the present case. In fact starting with the variational principle one may dispense with the heuristic argument used to single out the proper invariant distribution from many possibilities. The relation to topological entropy is also discussed. Note that the arguments are not confined to the case of Axiom A system.

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