Comments on the cosmic convergence of nonexpansive maps
Main results
We study the asymptotic behavior of nonexpansive maps using metric functionals, with applications to invariant subspaces and cosmic convergence.
1. Invariant subspaces of linear contractions
Let \(H\) be a real Hilbert space and \(T:H\to H\) an affine nonexpansive map of the form \(Tx=Ux+v\), where \(U\) is linear and \(\|U\|\leq1\).
If \(T\) has no approximate fixed points, that is, \(0\notin\overline{\operatorname{Im}(I-T)}\), then there exists a nonzero continuous linear functional \(f\) such that
\[ f(Tx)\geq f(x)\qquad(x\in H). \]Consequently, \(\ker f\) is a closed invariant hyperplane for \(U\).
2. Nonexpansive maps on \(\ell_1\)
Every nonexpansive map \(T:\ell_1(\mathbb Z)\to\ell_1(\mathbb Z)\) admits a nontrivial metric functional \(h\) satisfying
\[ h(Tx)\leq h(x)\qquad(x\in\ell_1(\mathbb Z)). \]Moreover, there exists a nonzero continuous linear functional \(f\) such that
\[ f(T^n0)\geq0\qquad(n\geq0). \]Thus, the entire orbit of the origin lies in a closed half-space.
3. Convergence of internal metric functionals without cosmic convergence
Consider the fixed-point-free nonexpansive map on \(\ell_1(\mathbb N)\),
\[ T(x_1,x_2,\ldots)=(1,x_1,x_2,\ldots). \]The internal metric functionals associated with the orbit \((T^n0)\) converge pointwise to
\[ h(x)=\sum_{i=1}^{\infty}\bigl(|x_i-1|-1\bigr), \]while the normalized iterates \(T^n0/\|T^n0\|_1\) have no weak cluster points.
This illustrates how metric functionals capture asymptotic behavior that cosmic convergence does not detect.
