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Let be an Anosov diffeomorphism whose linearization is irreducible. Assume that is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable subbundle and the unstable subbundle are jointly integrable, then is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization . Moreover, admits the finest dominated splitting in the weak stable subbundle with dimensions matching those for , and it has spectral rigidity along all these subbundles. In dimension 4, we also obtain a similar result by grouping the weak stable and unstable subbundles together as a center subbundle and assuming joint integrability of the strong stable and unstable subbundles. As an application, we show that for every symplectic diffeomorphism that is ‐close to an irreducible nonconformal automorphism , the extremal subbundles of are jointly integrable if and only if is smoothly conjugate to .

 

Polyethylene oxide (PEO)-based polymers are commonly studied for use as a solid polymer electrolyte for rechargeable Li-ion batteries; however, simultaneously achieving sufficient mechanical integrity and ionic conductivity has been a challenge. To address this problem, a customized polymer architecture is demonstrated wherein PEO bottle-brush arms are hyperbranched into a star architecture and then functionalized with end-grafted, linear PEO chains. The hierarchical architecture is designed to minimize crystallinity and therefore enhance ion transport via hyperbranching, while simultaneously addressing the need for mechanical integrity via the grafting of long, PEO chains ( M n = 10,000). The polymers are doped with lithium bis(trifluoromethane) sulfonimide (LiTFSI), creating hierarchically hyperbranched (HB) solid polymer electrolytes. Compared to electrolytes prepared with linear PEO of equivalent molecular weight, the HB PEO electrolytes increase the room temperature ionic conductivity from ∼2.5 × 10 –6 to 2.5 × 10 −5  S/cm. The conductivity increases by an additional 50% by increasing the block length of the linear PEO in the bottle brush arms from M n = 1,000 to 2,000. The mechanical properties are improved by end-grafting linear PEO ( M n = 10,000) onto the terminal groups of the HB PEO bottle-brush. Specifically, the Young’s modulus increases by two orders of magnitude to a level comparable to commercial PEO films, while only reducing the conductivity by 50% below the HB electrolyte without grafted PEO. This study addresses the trade-off between ion conductivity and mechanical properties, and shows that while significant improvements can be made to the mechanical properties with hierarchical grafting of long, linear chains, only modest gains are made in the room temperature conductivity.