부산대학교 도서관

Given a dynamical system $\phi\:X\to X$ on a topological space $X$, and a (numerical) function $h$ defined on $X$, the corresponding cohomological equation is the equation ${u\circ\phi-u=h}$, where the function $u$ is the unknown. Various problems in dynamical systems can be stated in terms of measuring the obstructions to solving a cohomological equation, in appropriate functional spaces. The paper under review studies the case of dynamical systems on $\Bbb{C}^2$ defined by linear automorphisms $A=\left(\smallmatrix a&1\\0&a\endsmallmatrix\right)$, with ${a\in\{\omega\in\Bbb{C}\,:\, |\omega|=1\}}$, in the functional space of holomorphic functions $\Cal{H}{\rm ol}(\Bbb{C}^2)$ (for linear automorphisms of $\Bbb{C}^2$, this is the only case which is not discussed in [A. El~Kacimi-Alaoui and T. Sohou, Proyecciones {\bf 22} (2003), no.~3, 243--271; MR2146077]). The authors prove that when $a$ is a root of unity, or $a$ is Diophantine, the unique obstructions to solving the corresponding cohomological equation on $\Cal{H}{\rm ol}(\Bbb{C}^2)$ are the obvious ones, in contrast to the case of Liouville $a$.