Transitive Maps Which Are Not Ergodic with Respect to Lebesgue Measure

In this note we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attrac-tors whose attractive basins arèintermingled', each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bi-modal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the New-ton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.