We evaluated some particular type of functional integral over the local gauge group C(R, U(1)) by going to a discretized lattice. The results explicitly violates the property of the Haar measure. We also analysed the FaddeevPopov method through a toy example. The results also violates the property of the Haar measure. PACS numbers:11.15.-q; 12.20.-m Typeset using REVTEX

Compact right topological groups arise in topological dynamics and in other settings. Following H. Furstenberg's seminal work on distal flows, R. Ellis and I. Namioka have shown that the compact right topological groups of dynamical type always admit a probability measure invariant under the continuous left translations; however, this invariance property is insufficient to identify a unique pro...

The Mahler formula gives an expression for the height of an algebraic number, as the integral of the log of the minimal equation with respect to the Haar measure on the circle. In the present work we prove that a similar result holds for nice self maps on the n-dimensional projective space. The height of the number is replaced by the canonical height of a hypersurface, and the Haar measure is r...

In this note we show that the periodic points of an expansive Zd action on a compact abelian group are uniformly distributed with respect to Haar measure if the action has completely positive entropy. In the general expansive case, we show that any measure obtained as the distribution of periodic points along some sequence of periods necessarily has maximal entropy but need not be Haar measure. §

We wish to thank Karen Rutherford, Erna Rollefson, Charles Hill, and Douglas MacPherson for assistance in various phases of these experiments, and Julius Adler for the Lederberg mutants and helpful comments. The following abbreviations are used: Gal, galactose; Gal-1-P, a-D-galactose-1-phosphate; G-1-P, a-D-glucose-l-phosphate; G-6-P, glucose-6-phosphate; UDPGal, uridine diphosphogalactose; UDP...