The displacement set of nonlinear maps in vector spaces

Given a vector space {V} and a map {f\colon V\to V} (linear or not), consider the displacement set of {f}, denoted {D(f) = \{f(x)-x\colon x\in V\}}. For linear maps this is simply the range of the operator {f-I} and therefore is a subspace.

The essentially nonlinear operations of taking the inverse or composition of maps become almost linear when the displacement set is considered. Specifically, if {f} has an inverse, then {D(f^{-1}) = -D(f)}, which is immediate from the definition. Also, {D(f\circ g)\subset D(f)+D(g)}.

When {V} is a topological vector space, the maps for which {D(f)} has compact closure are of particular interest: these are compact perturbations of the identity, for which degree theory can be developed. The consideration of {D(f)} makes it very clear that if {f} is an invertible compact perturbation of the identity, then {f^{-1}} is in this class as well.

It is also of interest to consider the maps for which {D(f)} is either bounded, or is bounded away from {0}. Neither case can occur for linear operators, so this is essentially nonlinear analysis. In the nonlinear case, the boundedness assumption for linear operators is usually replaced by the Lipschitz condition. Let us say that {f} is {(L, \ell)}-bi-Lipschitz if {\ell\|x-y\|\le \|f(x)-f(y)\|\le L\|x-y\|} for all {x, y} in the domain of {f}.

Brouwer’s fixed point theorem fails in infinite-dimensional Hilbert spaces, but it not yet clear how hard it can fail. The strongest possible counterexample would be a bi-Lipschitz automorphism of the unit ball with displacement bounded away from 0. The existence of such a map is unknown. If it does not exist, that would imply that the unit ball and the unit sphere in the Hilbert space are not bi-Lipschitz equivalent, because the unit sphere does have such an automorphism: {x\mapsto -x}.

Concerning the maps with bounded displacement, here is a theorem from Patrick Biermann’s thesis (Theorem 3.3.2): if {f} is an {(L, \ell)}-bi-Lipschitz map in a Hilbert space, {L/\ell < \pi/\sqrt{8}}, and {f} has bounded displacement, then {f} is onto. The importance of bounded displacement is illustrated by the forward shift map {S(x_1, x_2, \dots) = (0, x_1, x_2, \dots)} for which {L=\ell=1} but surjectivity nonetheless fails.

It would be nice to get rid of the assumption {L/\ell < \pi/\sqrt{8}} in the preceding paragraph. I guess any bi-Lipschitz map with bounded displacement should be surjective, at least in Hilbert spaces, but possibly in general Banach spaces as well.

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