If f : B 1 → B 1 is an extension of ϕ : S 0 → S 0 = { − 1, 1 } and ϕ has no fixed points, then f must have an interior fixed point, that is, a fixed point in int ( B 1 ).
However, if ϕ has a fixed point, then there need not be any interior fixed points.
We will prove that the interior fixed points do persist, even in this more general setting.
Model (2) has a unique interior fixed point ((u^, v^)) verifying (11).
If d ≥ 2 and f : B 2 → B 2 is a smooth extension of ϕ d, then f has at least one interior fixed point.
But it was proved in [1] (see also [2]) that if the extension f is smooth, it may still be required to have interior fixed points for certain maps ϕ that have many fixed points.
In Section 3, we prove that a smooth extension f : R 2 → R 2 of a power map ϕ d : S 1 → S 1 for d ≥ 2 must have at least one interior fixed point.
Schirmer generalized this interior fixed point result to smooth extensions f : B n → B n for n ≥ 2 to show in Example 4.7 of [3] that if f is a smooth extension of a 'sparse' map ϕ : S n − 1 → S n − 1, a generalization of ϕ d that is defined below, of degree d such that ( − 1 ) n d ≥ 2, then f must have at least one interior fixed point.
By the Julia-Carathédory theorem [13] (see also [7]) and the Wolff lemma [11], if (fin H mathbb{D,mathbb{D}})) with no interior fixed point, then there exists a unique regular boundary fixed point ξ such that (f'(xi in 0,1]); and if (fin H mathbb {D,mathbb{D}})) with an interior fixed point, then (f'(xi >1) for any boundary fixed point (xiinpartialmathbb{D}).
Of course the Brouwer fixed point theorem implies that a map f : B 2 → B 2 must have at least one interior fixed point if it is an extension of a map ϕ : S 1 → S 1 that has no fixed points.
