A complex Gap lemma

Biebler, Sébastien

doi:10.1090/proc/14716

A complex Gap lemma
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Proc. Amer. Math. Soc. 148 (2020), 351-364 Request permission

Abstract:

Inspired by the work of Newhouse in one real variable, we introduce a relevant notion of thickness for dynamical Cantor sets in the plane associated to a holomorphic IFS. Our main result is a complex version of Newhouse’s Gap Lemma: we show that under some assumptions, if the product of the thicknesses of two Cantor sets $K$ and $L$ is larger than 1, then $K$ and $L$ have non-empty intersection. Since in addition this thickness varies continuously, this gives a criterion to get a robust intersection between two Cantor sets in the plane.

References

Sebastien Biebler, Persistent homoclinic tangencies and infinitely many sinks for residual sets of automorphisms of low degree in $\mathbb {C}^3$, arXiv:1611.02011, 2016.
Gregery T. Buzzard, Stably interesting Julia sets of polynomials, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 11, 1013–1018 (English, with English and French summaries). MR 1249780
Gregery T. Buzzard, Infinitely many periodic attractors for holomorphic maps of $2$ variables, Ann. of Math. (2) 145 (1997), no. 2, 389–417. MR 1441881, DOI 10.2307/2951819
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Carlos Gustavo Moreira, There are no $C^1$-stable intersections of regular Cantor sets, Acta Math. 206 (2011), no. 2, 311–323. MR 2810854, DOI 10.1007/s11511-011-0064-0
Carlos Gustavo T. de A. Moreira and Jean-Christophe Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions, Ann. of Math. (2) 154 (2001), no. 1, 45–96. MR 1847588, DOI 10.2307/3062110
Sheldon E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18. MR 339291, DOI 10.1016/0040-9383(74)90034-2
Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641