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

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