Given a subvariety $V$ of the complex algebraic torus ${ \mathbb G}_{ \rm m}^n$ defined by polynomials of total degree at most $d$ and a power map $\phi: { \mathbb G}_{ \rm m}^n \to { \mathbb G}_{ \rm m}^n$, the points ${ \bf x}$ whose forward orbits ${ \mathcal O}_\phi({ \bf x})$ belong to $V$ form its { \em stable} subvariety $S(V,\phi)$. The main result of the paper provides an upper bound $T=T(n,d,\phi)$ for the number of iterations of the power map $\phi$ required to ``cut off'' the points of $V$ that do not belong to $S$.
Categories: math.NT math.DS