# A variation on Pisot–Vijayaraghavan numbers

Suppose a non-real algebraic integer $\alpha$ has, aside from itself and its complex conjugate $\bar\alpha$, all its algebraic conjugates of norm less than 1. Then the fractional parts of $\Re(\alpha^n)$ will cluster about 0, 1/2 and 1 as $n$ grows large.

Do such numbers have a name and/or a literature?

These numbers have been called complex Pisot numbers in at least a couple of papers. But they seem to have been investigated far less than their real counterparts.

Chamfy [1] and Garth [2] found the few smallest (in modulus) complex Pisot numbers.

Solomyak and Xu [3] showed that some of the dynamical properties of Pisot numbers continue to hold in the complex case, in particular, complex Bernoulli convolutions associated to complex Pisot numbers are singular. See Definition 2.2, Theorem 2.3 and the following remarks in [3].

References:

[1] Chamfy, Christiane. Fonctions méromorphes dans le cercle-unité et leurs séries de Taylor. (French) Ann. Inst. Fourier. Grenoble 8 1958 211--262.

[2] Garth, David. Complex Pisot numbers of small modulus. C. R. Math. Acad. Sci. Paris 336 (2003), no. 12, 967--970.

[3] Solomyak, Boris; Xu, Hui. On the `Mandelbrot set' for a pair of linear maps and complex Bernoulli convolutions. Nonlinearity 16 (2003), no. 5, 1733--1749.