Smale’s point estimate theory for Newton’s method on Lie groups

We introduce the notion of the (one-parameter subgroup) γ$\gamma$-condition for a map f$f$ from a Lie group to its Lie algebra and establish α$\alpha$-theory and γ$\gamma$-theory for Newton’s method for a map f$f$ satisfying this condition. Applications to analytic maps are provided, and Smale’s α$\alpha$-theory and γ$\gamma$-theory are extended and developed. Examples arising from initial value problems on Lie group are presented to illustrate applications of our results.