Given β > 1, let Tβ$\begin{array}{ccc}{\mathit{T}}_{\mathit{\beta }}\mathrm{:}\mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}& \mathrm{\to }& \mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}\\ \mathit{x}& \mathrm{\to }& \mathit{\beta x}\mathrm{-}\mathrm{\lfloor }\mathit{\beta x}\mathrm{\rfloor }\mathit{.}\end{array}$The iteration of this transformation gives rise to the greedy β-expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter β.In [17], K. Schmidt analyzed the set of periodic points of Tβ, where β is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansion$\begin{array}{ccc}\mathit{x}\mathrm{=}\sum _{\mathit{i}\mathrm{\ge }\mathrm{0}}{\mathit{e}}_{\mathit{i}}{\mathit{\beta }}^{\mathrm{-}\mathit{i}}\mathit{,}& & \end{array}$where each ei...

We use the theory of partially hyperbolic systems [HPS] in order to find singularities of index $1$ for vector fields with isolated zeroes in a $3$-ball. Indeed, we prove that such zeroes exists provided the maximal invariant set in the ball is partially hyperbolic, with volume expanding central subbundle, and the strong stable manifolds of the singularities are unknotted in the ball.