Lenticulary of the poles of the dynamical zeta function of the beta-shift in Lehmer's problem and limit Mahler measures

The dynamical zeta functions $\zeta_\beta (z)$ of the $\beta$-shift, for $\beta>1$ close to 1 being a reciprocal algebraic integer, give rise to a class of integer lacunary polynomials having moderate lacunarity controlled by an integer function, called the dynamical degree of $\beta$. The dynamical degree appears when the usual numeration system in base 10 is replaced by a variable numeration system in base $\beta$, and that the minimal polynomials are replaced by Parry Upper functions (which are generalized Fredholm determinants of some kind associated with $\zeta_\beta (z)$). This family of lacunary polynomials was never studied as such, except by Selmer (for heigth one trinomials) and Ljunggren (for quadrinomials). Results on their zero loci and factorizations are given in the context of general Theorems of Schinzel and Filaseta. Their asymptotic reducibility is studied by comparison with a variant Conjecture of Odlyzko Poonen’s Conjecture on Newman polynomials. The minoration of the Mahler measure, in the problem of Lehmer, arises from the existence of lenticuli of roots of these lacunary polynomials, in the cusp region of Solomyak’s fractal. For general (complex) algebraic integers $\alpha$, the lenticuli come from the dynamics of $\beta = \alpha$ and result in a Dobrowolski type minoration of the Mahler measure M ($\alpha$) as a function of the dynamical degree. The generalization to limit problems of rational points of algebraic varieties in higher dimension is discussed. In addition to the Boyd-Lawton Theorem, Condon’s asymptotic expansions, and Deninger’s cohomological approach, the present dynamical viewpoint applied to limit univariate Mahler measures enlights the role played by the dynamics of the houses of Mahler measures of algebraic numbers, and the inverse problem (Staines).