Orientation-preserving Transformations Research Articles

Gravitation and irreversibility

The internal time operatorM is an alternative to the usual dynamic time, an independent parameter of motion. Even when the dynamical entity is the three-geometry(3)G and we are concerned with its evolution in superspace (the problem of cosmological evolution), dynamical time remains an independent parameter associated with a choice of lapse and shift functions. The quantityM is, on the contrary, an ensemble-dependent parameter related to the “age” of a process: the entirety of the ensemble's evolution. With this different view of time as age, we seek a geometrical counterpart toM for the(3)G as an “ensemble.” For a closed negatively curved universe, a Lyapounov function can be established which allows anM to be defined for the Robertson-Walker universe. The time component of superspace momentumπ uv is introduced, and we identify its conjugate energy∂S/∂π uv with dissipation due to the evolving universe. A geometrical counterpart ofM is introduced by a conformal invariant Γ. This quantity simultaneously expresses (i) the topological feature of orientation-preserving transformations, and (ii) the Hamiltonian treatment of dissipative systems. This dual character of Γ, which links topological change to dissipative systems, suggests a geometrical basis forM. In this sense irreversibility is incorporated into the geometric structure of space-time, along with gravitation.

A Short Proof of the Cartwright-Littlewood Fixed Point Theorem

The purpose of this paper is to give a short proof of the Cartwright-Littlewood fixed point theorem (2, p. 3, Theorem A).Theorem A. If T is a (1-1) continuous and orientation preserving transformation of the Euclidean plane E onto itself which leaves a bounded continuum M invariant and if M does not separate E, then some point of M is left fixed by T.