Nil restricted Lie algebras of oscillating intermediate growth

The Grigorchuk and Gupta-Sidki groups are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2, Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic. It is known that the famous construction of Golod yields finitely generated associative nil-algebras of exponential growth. Recent extensions of that approach allowed to construct finitely generated associative nil-algebras of polynomial and intermediate growth. Another motivation of the paper is a construction of groups of oscillating growth by Kassabov and Pak. For any prime $p$ we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is “almost dying” by having a quasi-linear growth, namely the lower Gelfand-Kirillov dimension is one, more precisely, the growth is of type $n \big(\ln^{(q)} !n\big )^{\kappa}$\mathrm{.}Ontheotherhand,forinfinitelymany$n$ the growth has a rather fast intermediate behaviour of type $\exp (n\mathrm{/}(\ln n{)}^{\lambda })$, for such periods the algebra is “resuscitating”. Moreover, the growth function is oscillating between these two types of behaviour. These restricted Lie algebras have a nil $p$-mapping.