부산대학교 도서관

Given a dynamical system $\phi\:X\to X$ on a topological space $X$, anda (numerical) function $h$ defined on $X$, the correspondingcohomological equation is the equation ${u\circ\phi-u=h}$, where thefunction $u$ is the unknown. Various problems in dynamical systems canbe stated in terms of measuring the obstructions to solving acohomological equation, in appropriate functional spaces. The paperunder review studies the case of dynamical systems on $\Bbb{C}^2$defined by linear automorphisms $A=\left(\smallmatrixa&1\\0&a\endsmallmatrix\right)$, 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 tosolving the corresponding cohomological equation on $\Cal{H}{\rmol}(\Bbb{C}^2)$ are the obvious ones, in contrast to the case ofLiouville $a$.