Stable intersections of regular Cantor sets with large Hausdorff dimensions. (English) Zbl 1195.37015
Ann. Math. (2) 154, No. 1, 45-96 (2001); corrigendum ibid. 154, No. 2, 527 (2001); erratum ibid. 195, No. 1, 363-374 (2022).
Summary: We prove a conjecture by J. Palis [Differential equations, Proc. Lefschetz Centen. Conf., Mexico City/Mex. 1984, Pt. III, Contemp. Math. 58.3, 203–216 (1987; Zbl 0617.58027)] according to which the arithmetic difference of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval. More precisely, we prove that if the sum of the Hausdorff dimensions of two regular Cantor sets is bigger than one then, in almost all cases, there are translations of them whose intersection persistently has Hausdorff dimension.
MSC:
| 37C45 | Dimension theory of smooth dynamical systems |
| 28A78 | Hausdorff and packing measures |
| 28A80 | Fractals |
| 37D05 | Dynamical systems with hyperbolic orbits and sets |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
| 37G25 | Bifurcations connected with nontransversal intersection in dynamical systems |
