The pascal automorphism has a continuous spectrum

Abstract

Mathematicians who are only mathematicians have exact minds, provided all things are explained to them by means of definitions and axioms; otherwise they are inaccurate and insufferable, for they are only right when the principles are quite clear.Blaise Pascal, Thoughts, English translation by W. F. TrotterDedication. Dima Arnold (as well as me) was very fond of B. Pascal, and disliked R. Descartes, seeing him as a forerunner of Bourbakism he hated so much. As to me, in my youth I had great respect for Bourbaki, a very high opinion of his 5th volume, and even once wrote to him (N. Bourbaki) a long eulogistic letter, of which Dima did not approve. In reply, N. Bourbaki (impersonated by J. Dieudonné) presented me the next volume Integration, which had just appeared; the topic was close to my interests, but the volume turned out to be a failure. I was distressed and started to believe that perhaps Arnold was right.The keen interest to combinatorics and asymptotic problems, which appeared in the last years of V. I. Arnold's life and made us even closer, was, I believe, another manifestation of the fact that his mind revolted at any limits and prohibitions, he always violated canons, or, better to say, introduced new canons; and he was able to do this, because he was (according to Pascal) not only a mathematician.

Spectral theory of dynamical systems as diffraction theory of sampling functions

“…On the other hand, clearly, the condition that f continuous is stronger than f being bounded and measurable. We also note that a related result for subshifts over a finite alphabet can be found in Lemma 5 of [29]. There, discrete spectrum is characterized with a mean almost periodicity condition on points rather than functions.…”

The theory of filtrations of subalgebras, standardness, and independence

“…6 The theory of adic transformations and adic realizations of automorhisms became a new source of examples in ergodic theory. Even the first example suggested by the author, that of the Pascal automorphism, still intrigues researchers; it is not yet known whether its spectrum is continuous, and many other properties are also unknown (see [75], [79], [42], [24]).…”

Limiting Curves for Polynomial Adic Systems

“…where positive integers a j , k 1 (i), m j , are defined as in (4). Parameters N and l correspond to shifting the origin vertex (0, 0) to the vertex (N, l).…”