Invariant rigid geometric structures and expanding maps (Q2908166)

This work contributes to the program initiated by Zimmer and Gromov to classify chaotic differential dynamical systems preserving rigid geometric structures [\textit{G. D'ambra} and \textit{M. Gromov}, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 19--111 (1991; Zbl 0752.57017)]. The main result is Theorem 1.2, which states that if a smooth expanding map on a closed manifold preserves a topologically complete smooth rigid geometric structure, then it is smoothly conjugate to an expanding infranil endomorphism. The corollary to this result states that if a smooth expanding map on a closed manifold preserves a smooth generalized connection, then it is smoothly conjugate to an expanding infranil endomorphism.NEWLINENEWLINEThe theorem is proven by demonstrating several geometric propositions about locally homogeneous rigid geometric structures. After introducing elementary properties of rigid geometric structures in \S 2 and recalling the open-dense theorem of Gromov in \S 3, the author considers normal locally homogeneous structures in \S 4. Topological completeness and a notion of geodesic structure comprise \S 5. The preceding sections are then applied to expanding maps in \S 6 to prove the main theorem. Finally, based upon the topological completeness formulated in \S 5, which is a natural generalization of the classical completeness of linear connections, the author proves Corollary 1 in \S 7.