Arithmetic Difference of Middle Cantor Sets: Self-Similarity and Measure

Pourbarat, M.

doi:10.1080/1726037X.2012.10698612

Open Access Original Articles

Arithmetic Difference of Middle Cantor Sets: Self-Similarity and Measure

M. Pourbaratm-pourbarat@sbu.ac.irDept. of MathShahid Beheshti UniversityTehran, IranView full profile →

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pp. 107–114Vol. 10Issue 2November 2012DOI: 10.1080/1726037X.2012.10698612XML

Received:: 14 Mar 2011
Accepted:: 17 Jun 2012
Published Online:: 03 Jun 2013
Article type:: Original Articles
Language:: EN
Article no.:: 1726037X.2012.10698612
Pages:: 107–114

Abstract

Let C be the space of all middle Cantor sets, it is shown that for a dense subset L of C × C × ℝ, set Cα − λCβ forms a uniformly contracting self-similar set when (Cα, Cβ, λ) ∈ L. By selecting an appropriate element (CαCβ, λ) ∈ L, we see that Cα − λCβ does not satisfy open set condition and then we calculate its Lebesgue measure that is zero.

Keywords

Arithmetic difference iterated function system middle Cantor sets Palis conjecture

References

Freiman, G. A.1975 . , Kalink: Kalinin. Gosudarstv. Univ .
Hall, M.1947 . On the sum and product of continued fractions . , 48 : 966 – 993 .
Honary, B., Moreira, C. G. and Pourbarat, M.2005 . Stable intersections of affine Cantor sets . , 36 ( 3 ) : 363 – 378 .
Kan, I., Koak, H. and Yorke, J.1992 . Antimonotonicity: concurrent creation and annihilation of periodic orbits . , 2 : 219 – 252 .
Moreira, C. G. and Yoccoz, J.-C.2001 . Stable intersections of regular Cantor sets with large Hausdorff dimension . , 154 ( 1 ) : 45 – 96 .
Mendes, P. and Oliveira, F.1994 . On the topological structure of the arithmetic sum of two Cantor sets . , 7 ( 2 ) : 329 – 343 .
Newhouse, S.1979 . The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms . , 50 : 101 – 151 .
Pourbarat, M.Submitted
Palis, J. and Takens, F.1993 . , Cambridge: Cambridge Univ. Press .
Palis, J. and Yoccoz, J.-C.1997 . On the arithmetic sum of regular Cantor sets . , 14 ( 4 )
Sannami, A.1992 . An example of a regular Cantor set whose difference set is a Cantor set with positive measure . , 21 ( 1 ) : 7 – 24 .
Solomyak, B.1997 . On the arithmetic sums of Cantor sets . , : 133 – 141 .

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[1] Freiman, G. A.1975 . , Kalink: Kalinin. Gosudarstv. Univ .

[2] Hall, M.1947 . On the sum and product of continued fractions . , 48 : 966 – 993 .

[3] Honary, B., Moreira, C. G. and Pourbarat, M.2005 . Stable intersections of affine Cantor sets . , 36 ( 3 ) : 363 – 378 .

[4] Kan, I., Koak, H. and Yorke, J.1992 . Antimonotonicity: concurrent creation and annihilation of periodic orbits . , 2 : 219 – 252 .

[5] Moreira, C. G. and Yoccoz, J.-C.2001 . Stable intersections of regular Cantor sets with large Hausdorff dimension . , 154 ( 1 ) : 45 – 96 .

[6] Mendes, P. and Oliveira, F.1994 . On the topological structure of the arithmetic sum of two Cantor sets . , 7 ( 2 ) : 329 – 343 .

[7] Newhouse, S.1979 . The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms . , 50 : 101 – 151 .

[8] Pourbarat, M.Submitted

[9] Palis, J. and Takens, F.1993 . , Cambridge: Cambridge Univ. Press .

[10] Palis, J. and Yoccoz, J.-C.1997 . On the arithmetic sum of regular Cantor sets . , 14 ( 4 )

[11] Sannami, A.1992 . An example of a regular Cantor set whose difference set is a Cantor set with positive measure . , 21 ( 1 ) : 7 – 24 .

[12] Solomyak, B.1997 . On the arithmetic sums of Cantor sets . , : 133 – 141 .

Journal of Dynamical Systems and Geometric Theories

WoS JIF 2026 : 0.5 (Q3)

Arithmetic Difference of Middle Cantor Sets: Self-Similarity and Measure

Abstract

Keywords

References