The atomic-scale reduction mechanism of α-Fe2O3 nanowires by H2 was followed using transmission electron microscopy to reveal the evolution of atomic structures and the associated transformation pathways for different iron oxides. The reduction commences with the generation of oxygen vacancies that order onto every 10th (303false¯0) plane. This vacancy ordering is followed by an allotropic transformation of α-Fe2O3 → γ-Fe2O3 along with the formation of Fe3O4 nanoparticles on the surface of the γ-Fe2O3 nanowire by a topotactic transformation process, which shows 3D correspondence between the structures of the product and its host. These observations demonstrate that the partial reduction of α-Fe2O3 nanowires results in the formation of a unique hierarchical structure of hybrid oxides consisting of the parent oxide phase, γ-Fe2O3, as the one-dimensional wire and the Fe3O4 in the form of nanoparticles decorated on the parent oxide skeleton. We show that the proposed mechanism is consistent with previously published and our density functional theory results on the thermodynamics of surface termination and oxygen vacancy formation in α-Fe2O3. Compared to previous reports of α-Fe2O3 directly transformed to Fe3O4, our work provides a more in-depth understanding with substeps of reduction, i.e., the whole reduction process follows: α-Fe2O3 → α-Fe2O3 superlattice → γ-Fe2O3 + Fe3O4→ Fe3O4.

We consider a hyperbolic automorphism A : T 3 → T 3 of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold A into two oneparameter families of Anosov diffeomorphismsa conservative family and a dissipative one. For diffeomorphisms in these families we numerically calculate the strong unstable manifold of the fixed point. Our calculations strongly suggest that the strong unstable manifold is dense in T 3 . Further, we calculate push-forwards of the Lebesgue measure on a local strong unstable manifold. These numeric data indicate that the sequence of pushforwards converges to the SRB measure.