New examples of \(S\)-unimodal maps with a sigma-finite absolutely continuous invariant measure (Q945430)

\textit{S. D. Johnson} [Commun. Math. Phys. 110, 185--190 (1987; Zbl 0641.58024)] first showed the existence of non-renormalizable \(S\)-unimodal maps which do not have a finite absolutely continuous invariant measure (acim). Several authors asked if such maps could have a \(\sigma\)-finite acim. See \textit{F. Hofbauer} and \textit{G. Keller} [Commun. Math. Phys. 127, No. 2, 319--337 (1990; Zbl 0702.58034)], \textit{G. Keller} [Ergodic Theory Dyn. Syst. 10, No. 4, 717--744 (1990; Zbl 0715.58020)], and \textit{J. Guckenheimer} and \textit{S. Johnson} [Ann. Math. (2) 132, No. 1, 71--130 (1990; Zbl 0708.58007)]. Then \textit{F. Hofbauer} and \textit{G. Keller} [Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 4, 413--425 (1990; Zbl 0721.58018)] showed that \(S\)-unimodal maps without a finite acim could have an infinite \(\sigma\)-finite acim. Also \textit{H. Bruin} [Invariant Measures of Interval Maps, Ph.D. thesis, University of Delft (1994)] constructed maps without a finite acim but having a \(\sigma\)-finite acim such that the attractor is a certain interval. The paper reviewed here constructs new examples of \(S\)-unimodal maps which do not have a finite acim, but do have a \(\sigma\)-finite acim. The main result is that there are uncountably many maps in the quadratic family that admit no finite acim, but that have a \(\sigma\)-finite acim that is infinite on every interval. This is proven using Johnson boxes and the tower construction [see \textit{M. Jakobson}, Commun. Math. Phys. 81, 39--88 (1981; Zbl 0497.58017) and \textit{M. Jakobson} and \textit{G. Swiatek}, Ergodic Theory Dyn. Syst. 14, No. 4, 721--755 (1994; Zbl 0830.58019)]. The authors note that the property that the \(\sigma\)-finite acim is infinite on every interval was seen previously only for circle diffeomorphisms in \textit{Y. Katznelson} [J. Anal. Math. 31, 1--18 (1977; Zbl 0346.28012)]. Furthermore, the authors prove that any \(S\)-unimodal map has either a \(\sigma\)-finite acim or it has an invariant measure which is infinite on every set of positive Lebesgue measure.