# A gap principle for dynamics - Mathematics > Number Theory

Abstract: Let $f 1, .,f g\in {\mathbb C}z$ be rational functions, let$\Phi=f 1, .,f g$ denote their coordinatewise action on ${\mathbb P}^1^g$,let $V\subset {\mathbb P}^1^g$ be a proper subvariety, and let$P=x 1, .,x g\in {\mathbb P}^1^g{\mathbb C}$ be a nonpreperiodic pointfor $\Phi$. We show that if $V$ does not contain any periodic subvarieties ofpositive dimension, then the set of $n$ such that $\Phi^nP \in V{\mathbbC}$ must be very sparse. In particular, for any $k$ and any sufficiently large$N$, the number of $n \leq N$ such that $\Phi^nP \in V{\mathbb C}$ is lessthan $\log^k N$, where $\log^k$ denotes the $k$-th iterate of the $\log$function. This can be interpreted as an analog of the gap principle ofDavenport-Roth and Mumford.

Author: Robert L. Benedetto, Dragos Ghioca, Par Kurlberg, Thomas J. Tucker

Source: https://arxiv.org/