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September  2018, 38(9): 4259-4278. doi: 10.3934/dcds.2018186

On the arithmetic difference of middle Cantor sets

Mehdi Pourbarat

Dep. Math., Shahid Beheshti Univ., Tehran, Iran

Received  May 2017 Published  June 2018

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.

Keywords: Homoclinic bifurcations, Newhouse's Gap lemma, middle Cantor sets.
Mathematics Subject Classification: 28A78, 28A80, 58F14.
Citation: Mehdi Pourbarat. On the arithmetic difference of middle Cantor sets. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4259-4278. doi: 10.3934/dcds.2018186
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.

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.

Figure 1.  Blue region and curves determine all $\alpha, \beta$ that $C_\alpha+C_\beta= [0, ~2] $. For instant, the curves $r_{3, 1}, r_{2, 1}, r_{3, 2}, r_{4, 3}, r_{5, 4}$ are selected, that we characterized them by the functions $\beta=\alpha ^\frac{3}{1}, \alpha ^\frac{2}{1}, \alpha ^\frac{3}{2}, \alpha ^\frac{4}{3}, \alpha ^\frac{5}{4}$, respectively. The yellow Curves are the graph of these functions from downside to upside which have been drawn by Maple program
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Figure 2.  Gray region illustrates the recurrent set $R$
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Figure 3.  For middle Cantor sets, the sets $E, F, G, \Delta_1, \Delta_2$ and intervals $I, J$ are illustrated. Triangles $\Delta_1$ and $\Delta_2$ are non-empty and both project to the horizontal interval $I$
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Figure 4.  The graph of map $T$ is illustrated
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Figure 5.  The functions $g_0$ and $g_k$ are illustrated
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