# Average rotation

of the argument $\mathop{\rm arg} f(t)$ of a complex-valued, uniformly almost-periodic function $f(t)$

A phenomenon consisting of the existence (given certain conditions, see below) of the limit

$$\lim\limits _ {t \rightarrow \infty } \frac{ \mathop{\rm arg} f(t) }{t} = c.$$

The limit itself is also called the average rotation (mean motion). If $| f(t) | \neq 0$ for all $t$, then the selection of a continuous branch of $\mathop{\rm arg} f(t)$ is implied. For analytic almost-periodic functions, the concept of average rotation can be retained even when $f$ contains zeros. Namely, the concepts of "right" and "left" arguments are introduced, whose difference jumps by $\pm k \pi$ at a zero of multiplicity $k$ of $f$, and one therefore also speaks of right and left average rotation, unless they coincide, in which case one simply speaks of average rotation.

The question of the average rotation arose in connection with the fact that in celestial mechanics the longitude of the perihelion of a planet is expressed approximately as the argument of a certain trigonometric polynomial

$$f = \sum _ { j=1 } ^ { n } a _ {j} e ^ {i \omega _ {j} t } .$$

J.L. Lagrange studied two simple cases, namely, where one of the $| a _ {j} |$ is greater than the sum of the remaining coefficients, and for $n=2$; he noted that in other cases the question is complicated. The study of this question has been taken up only in the 20th century (for its history, see [1][3]). The final result that a trigonometric polynomial always has an average rotation was stated in 1938 by B. Jessen (for its proof, see [1]). (From the point of view of the theory of dynamical systems, it is a matter of averaging a certain function on a torus along the trajectories of the flow defined by shifts by elements of a one-parameter subgroup. However, this function has singularities that obstruct the automatic use of the corresponding general theorem.) Even earlier than this, H. Bohr proved the existence of an average rotation for any uniformly almost-periodic function for which $\inf | f(t) | > 0$( see [4]). In this case, the difference $\mathop{\rm arg} f(t)-ct$ is a uniformly almost-periodic function and is bounded. The average rotation of analytic almost-periodic functions in the general case has also been studied (see [1], [4]). In this case, the average rotation does not always exist, but if it does, then the difference $\mathop{\rm arg} f(t)-ct$ is not necessarily bounded. Nonetheless, it can still possess certain generalized properties of almost-periodicity; this is true, in particular, for trigonometric polynomials [5]. Except in the analytic case, only isolated results concerning the average rotation of a function $f$ for which $| f(t) | \neq 0$, $\inf | f(t) | = 0$, exist (see [6], [7]).

#### References

 [1] B. Jessen, H. Tornehave, "Mean motions and zeros of almost periodic functions" Acta Math. , 77 (1945) pp. 137–279 MR0015558 Zbl 0061.16504 [2] B. Jessen, "Some aspects of the theory of almost periodic functions" , Proc. Internat. Congress Mathematicians (Amsterdam, 1954) , 1 , North-Holland (1954) pp. 304–351 MR0095385 Zbl 0079.10402 [3] H. Weyl, "Mean motion" Amer. J. Math. , 60 (1938) pp. 889–896 MR1507355 Zbl 0019.33404 Zbl 64.0256.02 [4] B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian) MR0060629 Zbl 1222.42002 [5] R. Doss, "On mean motion" Amer. J. Math. , 79 : 2 (1957) pp. 389–396 MR0085390 Zbl 0077.08304 [6] B.M. Levitan, Mat. Zametki , 1 : 1 (1967) pp. 35–44 [7] E.A. Gorin, "A function algebra invariant of the Bohr–van Kampen theorem" Mat. Sb. , 82 (124) (1970) pp. 260–272 (In Russian)