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On the arithmetic difference of middle Cantor sets
Dep. Math., Shahid Beheshti Univ., Tehran, Iran |
We determine all triples $(α, β, λ)$ such that $C_α- λ C_β $ forms a closed interval, where $C_α$ and $C_β$ are middle Cantor sets. This follows from a new recurrence type result for certain renormalization operators. We also consider the affine Cantor sets $K$ and $K'$ defined by two increasing maps which the product of their thicknesses is bigger than one. Then we construct a recurrent set for their renormalization operators. This leads us to characterize all $λ$ that $K- λ K' $ is a closed interval.
References:
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G. Brown and W. Moran,
Raikov systems and radicals in convolution measure algebras, J. London Math. Soc., 28 (1983), 531-542.
doi: 10.1112/jlms/s2-28.3.531. |
[2] |
G. Brown, M. Keane, W. Moran and C. Pearce,
An inequality with applications to Cantor measures and normal numbers, Mathematika, 35 (1988), 87-94.
doi: 10.1112/S0025579300006306. |
[3] |
C. A. Cabrelli, K. E. Hare and U. M. Molter,
Sums of Cantor sets yielding an interval, J. Aust. Math. Soc., 73 (2002), 405-418.
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T. Cusick and M. Flahive,
The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1989.
doi: 10. 1090/surv/030. |
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Diophantine Approximation and the Geometry of Numbers (Markov's Problem), Kalinin. Gosudarstv. Univ., Kalink, 1975. |
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M. Hall,
On the sum and product of continued fractions, Ann. of Math., 48 (1947), 966-993.
doi: 10.2307/1969389. |
[7] |
B. Honary, C. G. Moreira and M. Pourbarat,
Stable intersections of affine Cantor sets, Bull. Braz. Math. Soc., 36 (2005), 363-378.
doi: 10.1007/s00574-005-0044-0. |
[8] |
K. Ilgar Eroglu,
On the arithmetic sums of Cantor sets, Nonlinearity, 20 (2007), 1145-1161.
doi: 10.1088/0951-7715/20/5/005. |
[9] |
P. Mendes and F. Oliveira,
On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity, 7 (1994), 329-343.
doi: 10.1088/0951-7715/7/2/002. |
[10] |
P. Mendes,
Sum of Cantor sets: Self-similarity and measure, Amer. Math. Soc., 127 (1999), 3305-3308.
doi: 10.1090/S0002-9939-99-05107-2. |
[11] |
C. G. Moreira,
Stable intersections of Cantor sets and homoclinic bifurcations, Ann. Inst. H. Poincare Anal. Non Lineaire, 13 (1996), 741-781.
doi: 10.1016/S0294-1449(16)30122-6. |
[12] |
C. G. Moreira and J.-C. Yoccoz,
Stable intersections of regular Cantor sets with large Hausdorff dimension, Ann. of Math., 154 (2001), 45-96.
doi: 10.2307/3062110. |
[13] |
S. Newhouse,
The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. IHES, 50 (1979), 101-151.
|
[14] |
S. Newhouse,
Non density of Axiom A(a) on $S^2$, Proc. A.M.S. Symp. Pure Math., 14 (1970), 191-202.
|
[15] |
S. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
[16] |
J. Palis and F. Takens,
Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Univ. Press, Cambridge, 1993. |
[17] |
J. Palis and J.-C. Yoccoz,
On the arithmetic sum of regular Cantor sets, Ann. Inst. Henri Poincare, 14 (1997), 439-456.
doi: 10.1016/S0294-1449(97)80135-7. |
[18] |
Y. Peres and P. Shmerkin,
Resonance between Cantor sets, Ergod. Th. and Dynam. Sys., 29 (2009), 201-221.
doi: 10.1017/S0143385708000369. |
[19] |
M. Pourbarat, Stable intersection of middle-$α$ Cantor sets Comm. in Cont. Math., 17 (2015), 1550030, 19 pp.
doi: 10. 1142/S0219199715500303. |
[20] |
A. Sannami,
An example of a regular Cantor set whose difference set is a Cantor set with positive measure, Hokkaido Math. J., 21 (1992), 7-24.
doi: 10.14492/hokmj/1381413267. |
[21] |
B. Solomyak,
On the arithmetic sums of Cantor sets, Indag. Mathem., 8 (1997), 133-141.
doi: 10.1016/S0019-3577(97)83357-5. |
show all references
References:
[1] |
G. Brown and W. Moran,
Raikov systems and radicals in convolution measure algebras, J. London Math. Soc., 28 (1983), 531-542.
doi: 10.1112/jlms/s2-28.3.531. |
[2] |
G. Brown, M. Keane, W. Moran and C. Pearce,
An inequality with applications to Cantor measures and normal numbers, Mathematika, 35 (1988), 87-94.
doi: 10.1112/S0025579300006306. |
[3] |
C. A. Cabrelli, K. E. Hare and U. M. Molter,
Sums of Cantor sets yielding an interval, J. Aust. Math. Soc., 73 (2002), 405-418.
doi: 10.1017/S1446788700009058. |
[4] |
T. Cusick and M. Flahive,
The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1989.
doi: 10. 1090/surv/030. |
[5] |
G. A. Freiman,
Diophantine Approximation and the Geometry of Numbers (Markov's Problem), Kalinin. Gosudarstv. Univ., Kalink, 1975. |
[6] |
M. Hall,
On the sum and product of continued fractions, Ann. of Math., 48 (1947), 966-993.
doi: 10.2307/1969389. |
[7] |
B. Honary, C. G. Moreira and M. Pourbarat,
Stable intersections of affine Cantor sets, Bull. Braz. Math. Soc., 36 (2005), 363-378.
doi: 10.1007/s00574-005-0044-0. |
[8] |
K. Ilgar Eroglu,
On the arithmetic sums of Cantor sets, Nonlinearity, 20 (2007), 1145-1161.
doi: 10.1088/0951-7715/20/5/005. |
[9] |
P. Mendes and F. Oliveira,
On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity, 7 (1994), 329-343.
doi: 10.1088/0951-7715/7/2/002. |
[10] |
P. Mendes,
Sum of Cantor sets: Self-similarity and measure, Amer. Math. Soc., 127 (1999), 3305-3308.
doi: 10.1090/S0002-9939-99-05107-2. |
[11] |
C. G. Moreira,
Stable intersections of Cantor sets and homoclinic bifurcations, Ann. Inst. H. Poincare Anal. Non Lineaire, 13 (1996), 741-781.
doi: 10.1016/S0294-1449(16)30122-6. |
[12] |
C. G. Moreira and J.-C. Yoccoz,
Stable intersections of regular Cantor sets with large Hausdorff dimension, Ann. of Math., 154 (2001), 45-96.
doi: 10.2307/3062110. |
[13] |
S. Newhouse,
The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. IHES, 50 (1979), 101-151.
|
[14] |
S. Newhouse,
Non density of Axiom A(a) on $S^2$, Proc. A.M.S. Symp. Pure Math., 14 (1970), 191-202.
|
[15] |
S. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
[16] |
J. Palis and F. Takens,
Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Univ. Press, Cambridge, 1993. |
[17] |
J. Palis and J.-C. Yoccoz,
On the arithmetic sum of regular Cantor sets, Ann. Inst. Henri Poincare, 14 (1997), 439-456.
doi: 10.1016/S0294-1449(97)80135-7. |
[18] |
Y. Peres and P. Shmerkin,
Resonance between Cantor sets, Ergod. Th. and Dynam. Sys., 29 (2009), 201-221.
doi: 10.1017/S0143385708000369. |
[19] |
M. Pourbarat, Stable intersection of middle-$α$ Cantor sets Comm. in Cont. Math., 17 (2015), 1550030, 19 pp.
doi: 10. 1142/S0219199715500303. |
[20] |
A. Sannami,
An example of a regular Cantor set whose difference set is a Cantor set with positive measure, Hokkaido Math. J., 21 (1992), 7-24.
doi: 10.14492/hokmj/1381413267. |
[21] |
B. Solomyak,
On the arithmetic sums of Cantor sets, Indag. Mathem., 8 (1997), 133-141.
doi: 10.1016/S0019-3577(97)83357-5. |




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