enough, whereas if the $E_i$ are Lipschitz and defined on (nontrivial) closed arcs then no assumption on $X$ is needed. “Intermediate" cases involve the [type]{} of the Banach space $X$ and H" older conditions on the $E_i$. What Corollary \[eigvectfield\] says is that continuity of the $E_i$ is *always* enough, regardless of the underlying space $X$.

We also point out that the assumption in (i), i.e. the existence of $\TT$-eigenvector fields with the required spanning property defined on perfect sets, is in fact equivalent to the perfect spanning property ([@Sophie]). Likewise, the assumption in (ii) is equivalent to the $\mathcal U_0$-perfect spanning property (see Proposition \[perfect\]).

In order to illustrate our results, two examples are worth presenting immediately. Other examples will be reviewed in section \[final\].

Let $\mathbf w=(w_n)_{n\geq 1}$ be a bounded sequence of nonzero complex numbers, and let $B_{\bf w}$ be the associated *weighted backward shift* acting on $X_p=\ell^p(\NN)$, $1\leq p<\infty$ or $X_\infty=c_0(\NN)$; that is, $B_{\bf w}(x_0,x_1,x_2,\dots )=(w_1x_1,w_2x_2,\dots ).$ Solving the equation $B_{\mathbf w}(x)=\lambda x$, it is easy to check that $B_{\mathbf w}$ has eigenvalues of modulus 1 if and only if $$\label{shift} 
\hbox{the sequence $\displaystyle\left(\frac{1}{w_0\cdots w_n}\right)_{n\geq 0}$ is in $X_p$}$$ (we have put $w_0:=1$). In this case 