Dynamics of translations on maximal compact subgroups

The paper rigorously proves that hyperbolic translations on K ≃ G/AN are gradient for a specifically constructed K-invariant “Borel” metric and characterizes fixed-point components, stable manifolds, recurrent/chain-recurrent sets, Morse components, and normal hyperbolicity using the Jordan decomposition; see Theorem 1.1 and Theorem 1.2 and their proofs, including the metric construction and stable-set descriptions (e.g., fix(H,u)=K^0_H u b; st(H,u)=N^-_H K^0_H u b; attractors indexed by C up to C_H; and st(g^t,u)=N^-_H M(g^t,u)) . The candidate solution reproduces most end results but its core gradient argument is incorrect: it attempts to use the Iwasawa projection Φ_H(k)=⟨H,ℋ(k)⟩ on K; however ℋ(k)=0 for k∈K, so Φ_H is constant and cannot generate the flow. Moreover, it misapplies DKV’s flag-manifold gradient result to K and conflates K with K/M. The paper instead builds the correct K-invariant metric on K to obtain the gradient property and then derives the rest (including normal hyperbolicity with exponential domination over the unipotent’s polynomial growth) . Minor model imprecisions also appear in attractor indexing (C vs C_H\C) and in the stated growth bounds for normal hyperbolicity.