Intricate Structure of the Analyticity Set for Solutions of a Class of Integral Equations

We consider a class of compact positive operators \(L:X\rightarrow X\) given by \((Lx)(t)=\int ^t_{\eta (t)}x(s)\,ds\), acting on the space X of continuous \(2\pi \)-periodic functions x. Here \(\eta \) is continuous with \(\eta (t)\le t\) and \(\eta (t+2\pi )=\eta (t)+2\pi \) for all \(t\in \mathbf{R}\). We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem \(\kappa x=Lx\) exists for some \(\kappa >0\) (equal to the spectral radius of L) by the Krein–Rutman theorem. If additionally \(\eta \) is analytic, we study the set \({\mathcal {A}}\subseteq \mathbf{R}\) of points t at which x is analytic; in general \({\mathcal {A}}\) is a proper subset of \(\mathbf{R}\), although x is \(C^\infty \) everywhere. Among other results, we obtain conditions under which the complement \({\mathcal {N}}=\mathbf{R}{\setminus }{\mathcal {A}}\) of \({\mathcal {A}}\) is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map \(t\rightarrow \eta (t)\).