American Institute of Mathematical Sciences

Advanced
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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Gray region illustrates the recurrent set $R$
Figure Options

Download full-size image

Download as PowerPoint slide

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 $I$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The graph of map $T$ is illustrated
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The functions $g_0$ and $g_k$ are illustrated
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Enrique R. Pujals. Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 335-405. doi: 10.3934/dcds.2008.20.335
[2]

S. Astels. Thickness measures for Cantor sets. Electronic Research Announcements, 1999, 5: 108-111.

[3]

Yannan Liu, Linfen Cao. Lifespan theorem and gap lemma for the globally constrained Willmore flow. Communications on Pure and Applied Analysis, 2014, 13 (2) : 715-728. doi: 10.3934/cpaa.2014.13.715
[4]

Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375
[5]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[6]

Silvére Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik. Computability of topological entropy: From general systems to transformations on Cantor sets and the interval. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4259-4286. doi: 10.3934/dcds.2020180
[7]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[8]

Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693
[9]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133
[10]

S. Bautista, C. Morales, M. J. Pacifico. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 761-775. doi: 10.3934/dcds.2007.19.761
[11]

Panos K. Palamides, Alex P. Palamides. Singular boundary value problems, via Sperner's lemma. Conference Publications, 2007, 2007 (Special) : 814-823. doi: 10.3934/proc.2007.2007.814
[12]

Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063
[13]

Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179
[14]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021
[15]

Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069
[16]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2677-2698. doi: 10.3934/dcds.2020381
[17]

V. Afraimovich, T.R. Young. Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 691-704. doi: 10.3934/dcds.2000.6.691
[18]

Wenli Ren, Feng Wang. A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020131
[19]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293
[20]

Thorsten Riess. Numerical study of secondary heteroclinic bifurcations near non-reversible homoclinic snaking. Conference Publications, 2011, 2011 (Special) : 1244-1253. doi: 10.3934/proc.2011.2011.1244

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (291)
  • HTML views (157)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy