Non-autonomous basins with uniform bounds are elliptic (Q2821732)

The article focusses on the so-called Bedford conjecture about the biholomorphic structure of the non-autonomous basin of attraction of a system \((f_j)_{j\in\mathbb N}\) of holomorphic automorphisms of \(\mathbb C^n\) with the following bounds: For some \(0 < a < b <1\) the inequality \(a|z|\leq |f_j(z)|\leq b|z|\) is fulfilled for all \(j\in\mathbb N\) and all \(z\in\mathbb B^n\). The conjecture claims that NEWLINENEWLINE\[NEWLINE\Omega:=\{z\in \mathbb C^n\,\big|\,\lim_{j\rightarrow\infty} f_j\circ f_{j-1}\circ\dots\circ f_1(z)=0\}NEWLINE\]NEWLINE NEWLINEis biholomorphically equivalent to \(\mathbb C^n\). Several weaker versions of the conjecture have been verified, see the survey article by \textit{A. Abbondandolo} et al. [``A survey on non-autonomous basins in several complex variables'', \url{arxiv:1311.3835}]. The main result of the paper under review states that \(\Omega\) is elliptic in the sense of Gromov: it admits a dominating spray \(s\), i.e., a holomorphic map \(s:E\rightarrow\Omega\) from the total space of a holomorphic vector bundle \(E\) over \(\Omega\) with \(s(0_z)=z\) and \(s|_{E_z}\rightarrow\Omega\) a submersion at \(0_z\) for all \(z\in\Omega\). The trivial vector bundle \(\mathbb C^n\times\Omega\rightarrow\Omega\) carries a dominating holomorphic spray which is obtained as the limit of an ascending sequence of sprays \(s_m:K_m\times m\mathbb B^n\rightarrow\Omega\) with respect to the compact exhaustion \(K_1\subset\cdots\subset K_m:=(f_m\circ\dots\circ f_1)^{-1}(\overline{\mathbb B^n})\subset\cdots\) of \(\Omega\). Since elliptic manifolds are Oka, any holomorphic map \(K\rightarrow\Omega\) of a compact convex subset \(K\) of a Stein manifold \(Y\) can be uniformly approximated on \(K\) by holomorphic maps \(Y\rightarrow\Omega\), see [\textit{F. Forstnerič}, Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis. Berlin: Springer (2011; Zbl 1247.32001)]. The authors use this fact for the construction of a holomorphic map \(f:Y\rightarrow\Omega\) with dense image. The construction process can be modified to yield an embedding \(Y\rightarrow\Omega\) when \(2\dim Y+1\leq n\) and a surjection \(Y\rightarrow\Omega\) when \(\dim Y\geq n\). It is interesting to realize that the weaker condition \(0\leq a < b < 1\) would not be sufficient for \(\Omega \) to be biholomorphically equivalent to \(\mathbb C^n\). The first named author gave an example where \(\Omega\) carries a non-constant bounded plurisubharmonic function in [Adv. Stud. Pure Math. 42, 95--108 (2004; Zbl 1071.32007)].