Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-17T02:23:11.191Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological structure of the sum of two homogeneous Cantor sets

Part of: Classical measure theory Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  24 March 2022

MEHDI POURBARAT
MEHDI POURBARAT*
Affiliation:
Department of Mathematics, Shahid Beheshti University, Tehran, Iran
*
e-mail: m-pourbarat@sbu.ac.ir
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that in the context of homogeneous Cantor sets, there are generically five possible (open and dense) structures for their arithmetic sum: a Cantor set, an L, R, M-Cantorval and a finite union of closed intervals. The dense case has been dealt with previously. In this paper, we explicitly present pairs of this space which have stable intersection, while not satisfying the generalized thickness test. Also, all the pairs of middle homogeneous Cantor sets whose arithmetic sum is a closed interval are identified.

Keywords

homogeneous Cantor setsPalis conjectureNewhouse’s thickness

MSC classification

Primary: 28A80: Fractals 28A78: Hausdorff and packing measures
Secondary: 37G25: Bifurcations connected with nontransversal intersection
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 5 , May 2023 , pp. 1712 - 1736
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Dedicated to Carlos Gustavo Moreira

References

Anisca, R. and Chlebovec, C.. On the structure of arithmetic sums of Cantor sets with constant ratios of dissection. Nonlinearity 22 (2009), 2127–2140.CrossRefGoogle Scholar
Anisca, R., Chlebovec, C. and Ilie, M.. The structure of arithmetic sums of affine Cantor sets. Real Anal. Exchange 37(2) (2012), 325–332.CrossRefGoogle Scholar
Anisca, R. and Ilie, M.. A technique of studying sums of central Cantor sets. Canad. Math. Bull. 44(1) (2001), 12–18.CrossRefGoogle Scholar
Cabrelli, C. A., Hare, K. E. and Molter, U. M.. Sums of Cantor sets yielding an interval. J. Aust. Math. Soc. 73(3) (2002), 405–418.CrossRefGoogle Scholar
Honary, B., Moreira, C. G. and Pourbarat, M.. Stable intersections of affine Cantor sets. Bull. Braz. Math. Soc. (N.S.) 36(3) (2005), 363–378.CrossRefGoogle Scholar
Ilgar Eroglu, K.. On the arithmetic sums of Cantor sets. Nonlinearity 20(5) (2007), 1145–1161.CrossRefGoogle Scholar
Mendes, P.. Sum of Cantor sets: self-similarity and measure. Amer. Math. Soc. 127(11) (1999), 3305–3308.CrossRefGoogle Scholar
Mendes, P. and Oliveira, F.. On the topological structure of the arithmetic sum of two Cantor sets. Nonlinearity 7(2) (1994), 329–343.CrossRefGoogle Scholar
Moreira, C. G.. Stable intersections of Cantor sets and homoclinic bifurcations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(6) (1996), 741–781.CrossRefGoogle Scholar
Moreira, C. G., Morales, E. M. and Letelier, J. R.. On the topology of arithmetic sums of regular Cantor sets. Nonlinearity 13 (2000), 2077–2087.CrossRefGoogle Scholar
Moreira, C. G. and Yoccoz, J.-C.. Stable intersections of regular Cantor sets with large Hausdorff dimension. Ann. of Math. (2) 154(1) (2001), 45–96.CrossRefGoogle Scholar
Newhouse, S.. Non density of Axiom A(a) on ${S}^2$ . Proc. Sympos. Pure Math. 14 (1970), 191–202.CrossRefGoogle Scholar
Newhouse, S.. Diffeomorphisms with infinitely many sinks. Topology 13 (1974), 9–18.CrossRefGoogle Scholar
Newhouse, S.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101–151.CrossRefGoogle Scholar
Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, 1993.Google Scholar
Peres, Y. and Shmerkin, P.. Resonance between Cantor sets. Ergod. Th. & Dynam. Sys. 29 (2009), 201–221.CrossRefGoogle Scholar
Pourbarat, M.. Stable intersection of middle- $\alpha$ Cantor sets. Commun. Contemp. Math. 17 (2015), 1550030.CrossRefGoogle Scholar
Pourbarat, M.. On the arithmetic difference of middle Cantor sets. Discrete Contin. Dyn. Syst. 38 (2018), 4259–4278.CrossRefGoogle Scholar
Pourbarat, M.. Stable intersection of affine Cantor sets defined by two expanding maps. J. Differential Equations 266 (2018), 2125–2141.CrossRefGoogle Scholar
Pourbarat, M.. On the arithmetic sum and product of middle Cantor sets, to appear.Google Scholar
Solomyak, B.. On the arithmetic sums of Cantor sets. Indag. Math. (N.S.) 8 (1997), 133–141.CrossRefGoogle Scholar