Annular itineraries for entire functions

In order to analyse the way in which the size of the iterates $(f^n(z))$ of a transcendental entire function $f$ can behave, we introduce the concept of the {\it annular itinerary} of a point $z$. This is the sequence of non-negative integers $s_0{s}_1\mathrm{.}\mathrm{.}\mathrm{.}$ defined by $$f^n(z)\in A_{s_n}(R),\text{for}n\ge 0,$$ where $A_0(R)=\{z:\mathrm{\mid }z\mathrm{\mid }<R\}$ and $$A_n(R)=\{z:M^{n-1}(R)\le \mathrm{\mid }z\mathrm{\mid }<M^n(R)\},n\ge 1.$$ Here $M(r)$ is the maximum modulus of $f$ and $R>0$ is so large that $M(r)>r$, for $r\ge R$. We consider the different types of annular itineraries that can occur for any transcendental entire function $f$ and show that it is always possible to find points with various types of prescribed annular itineraries. The proofs use two new annuli covering results that are of wider interest.