Journal of the
Korean Mathematical Society JKMS

Let $F: \mathbb C^k \rightarrow \mathbb C^k$ be a dynamical system and let $\{x_n\}_{n \geq 0}$ denote an orbit of $F.$ We study the relation between $\{x_n\}$ and pseudoorbits $\{y_n\}, y_0=x_0.$ Here $y_{n+1}=F(y_n)+s_n.$ In general $y_n$ might diverge away from $x_n.$ Our main problem is whether there exists arbitrarily small $t_n$ so that if $\tilde{y}_{n+1}=F(\tilde{y}_n)+s_n+t_n,$ then $\tilde{y}_n$ remains close to $x_n.$ This leads naturally to the concept of sustainable orbits, and their existence seems to be closely related to the concept of hyperbolicity, although they are not in general equivalent.

Keywords: sustainability, complex dynamics, hyperbolicity, H\'enon maps

MSC numbers: Primary 32H50; Secondary 37A25