Random intersections of thick Cantor sets
Author:
Roger L. Kraft
Journal:
Trans. Amer. Math. Soc. 352 (2000), 13151328
MSC (1991):
Primary 28A80; Secondary 58F99
Published electronically:
September 20, 1999
MathSciNet review:
1653359
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let , be Cantor sets embedded in the real line, and let , be their respective thicknesses. If , then it is well known that the difference set is a disjoint union of closed intervals. B. Williams showed that for some , it may be that is as small as a single point. However, the author previously showed that generically, the other extreme is true; contains a Cantor set for all in a generic subset of . This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if , then contains a Cantor set for almost all in .
 [GH]
John
Guckenheimer and Philip
Holmes, Nonlinear oscillations, dynamical systems, and bifurcations
of vector fields, Applied Mathematical Sciences, vol. 42,
SpringerVerlag, New York, 1990. Revised and corrected reprint of the 1983
original. MR
1139515 (93e:58046)
 [HKY]
Brian
R. Hunt, Ittai
Kan, and James
A. Yorke, When Cantor sets intersect
thickly, Trans. Amer. Math. Soc.
339 (1993), no. 2,
869–888. MR 1117219
(94f:28010), http://dx.doi.org/10.1090/S00029947199311172198
 [KP]
Richard
Kenyon and Yuval
Peres, Intersecting random translates of invariant Cantor
sets, Invent. Math. 104 (1991), no. 3,
601–629. MR 1106751
(92g:28018), http://dx.doi.org/10.1007/BF01245092
 [K1]
Roger
Kraft, Intersections of thick Cantor sets, Mem. Amer. Math.
Soc. 97 (1992), no. 468, vi+119. MR 1106988
(92i:28010), http://dx.doi.org/10.1090/memo/0468
 [K2]
Roger
L. Kraft, One point intersections of middle𝛼 Cantor
sets, Ergodic Theory Dynam. Systems 14 (1994),
no. 3, 537–549. MR 1293407
(95i:54050), http://dx.doi.org/10.1017/S0143385700008014
 [K3]
Roger
L. Kraft, What’s the difference between Cantor sets?,
Amer. Math. Monthly 101 (1994), no. 7, 640–650.
MR
1289273 (95f:04006), http://dx.doi.org/10.2307/2974692
 [K4]
, A golden Cantor set, Amer. Math. Monthly 105 (8) (1998). CMP 99:01
 [KKY]
Ittai
Kan, Hüseyin
Koçak, and James
A. Yorke, Antimonotonicity: concurrent creation and annihilation of
periodic orbits, Ann. of Math. (2) 136 (1992),
no. 2, 219–252. MR 1185119
(94c:58135), http://dx.doi.org/10.2307/2946605
 [MO]
Pedro
Mendes and Fernando
Oliveira, On the topological structure of the arithmetic sum of two
Cantor sets, Nonlinearity 7 (1994), no. 2,
329–343. MR 1267692
(95j:58123)
 [N1]
Sheldon
E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth
stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ.
Math. 50 (1979), 101–151. MR 556584
(82e:58067)
 [N2]
Sheldon
E. Newhouse, Lectures on dynamical systems, Dynamical systems
(C.I.M.E. Summer School, Bressanone, 1978) Progr. Math., vol. 8,
Birkhäuser, Boston, Mass., 1980, pp. 1–114. MR 589590
(81m:58028)
 [PT1]
J.
Palis and F.
Takens, Hyperbolicity and the creation of homoclinic orbits,
Ann. of Math. (2) 125 (1987), no. 2, 337–374.
MR 881272
(89b:58118), http://dx.doi.org/10.2307/1971313
 [PT2]
Jacob
Palis and Floris
Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic
bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35,
Cambridge University Press, Cambridge, 1993. Fractal dimensions and
infinitely many attractors. MR 1237641
(94h:58129)
 [PS]
Yuval
Peres and Boris
Solomyak, Selfsimilar measures and
intersections of Cantor sets, Trans. Amer.
Math. Soc. 350 (1998), no. 10, 4065–4087. MR 1491873
(98m:26009), http://dx.doi.org/10.1090/S0002994798022922
 [R]
David
Ruelle, Elements of differentiable dynamics and bifurcation
theory, Academic Press, Inc., Boston, MA, 1989. MR 982930
(90f:58048)
 [S]
Atsuro
Sannami, An example of a regular Cantor set whose difference set is
a Cantor set with positive measure, Hokkaido Math. J.
21 (1992), no. 1, 7–24. MR 1153749
(93c:58116), http://dx.doi.org/10.14492/hokmj/1381413267
 [W]
R.
F. Williams, How big is the intersection of two thick Cantor
sets?, Continuum theory and dynamical systems (Arcata, CA, 1989)
Contemp. Math., vol. 117, Amer. Math. Soc., Providence, RI, 1991,
pp. 163–175. MR 1112813
(92f:58116), http://dx.doi.org/10.1090/conm/117/1112813
 [WZ]
Richard
L. Wheeden and Antoni
Zygmund, Measure and integral, Marcel Dekker, Inc., New
YorkBasel, 1977. An introduction to real analysis; Pure and Applied
Mathematics, Vol. 43. MR 0492146
(58 #11295)
 [GH]
 J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, SpringerVerlag, New York, 1983. MR 93e:58046
 [HKY]
 B.R. Hunt, I. Kan, J. A. Yorke, When Cantor sets intersect thickly, Trans. Amer. Math Soc. 339 (2) (1993), 869888. MR 94f:28010
 [KP]
 R. Kenyon, Y. Peres, Intersecting random translates of invariant Cantor sets, Invent. Math. 104 (3) (1991), 601629. MR 92g:28018
 [K1]
 R. L. Kraft, Intersections of Thick Cantor Sets, Mem. Amer. Math. Soc. 97 (468) (1992). MR 92i:28010
 [K2]
 , One point intersections of middle Cantor sets, Ergodic Theory Dynam. Systems 14 (3) (1994), 537549. MR 95i:54050
 [K3]
 , What's the difference between Cantor sets, Amer. Math. Monthly 101 (7) (1994), 640650. MR 95f:04006
 [K4]
 , A golden Cantor set, Amer. Math. Monthly 105 (8) (1998). CMP 99:01
 [KKY]
 I. Kan, H. Ko[??]ak, J. Yorke, Antimonotonicity: concurrent creation and annihilation of periodic orbits, Ann. of Math. (2) 136 (2) (1992), 219252. MR 94c:58135
 [MO]
 P. Mendes, F. Oliveira, On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity 7 (2) (1994), 329343. MR 95j:58123
 [N1]
 S. E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. IHES 50 (1979), 101151. MR 82e:58067
 [N2]
 , Lectures on dynamical systems, Dynamical Systems, C. I. M. E. Lectures, Bressanone, Italy, June, 1978, Progress in Mathematics, No. 8, Birkhäuser, Boston, 1980, pp. 1114. MR 81m:58028
 [PT1]
 J. Palis, F. Takens, Hyperbolicity and the creation of homoclinic orbits, Ann. of Math. (2) 125 (2) (1987), 337374. MR 89b:58118
 [PT2]
 , Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: Fractal dimensions and infinitely many attractors, Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993. MR 94h:58129
 [PS]
 Y. Peres, B. Solomyak, Selfsimilar measures and intersections of Cantor sets, Trans. Amer. Math Soc. 350 (10) (1998), 40654087. MR 98m:26009
 [R]
 D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, New York, 1989. MR 90f:58048
 [S]
 A. Sannami, An example of a regular Cantor set whose difference set is a Cantor set with positive measure, Hokkaido Math. J. 21 (1) (1992), 724. MR 93c:58116
 [W]
 R. F. Williams, How big is the intersection of two thick Cantor sets?, Continuum Theory and Dynamical Systems (M. Brown, ed.), Proc. Joint Summer Research Conference on Continua and Dynamics (Arcata, California, 1989), Amer. Math. Soc., Providence, R.I., 1991. MR 92f:58116
 [WZ]
 R. Wheeden, A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Marcel Dekker, Inc., New York, 1977. MR 58:11295
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
28A80,
58F99
Retrieve articles in all journals
with MSC (1991):
28A80,
58F99
Additional Information
Roger L. Kraft
Affiliation:
Department of Mathematics, Computer Science and Statistics, Purdue University Calumet, Hammond, Indiana 46323
Email:
roger@calumet.purdue.edu
DOI:
http://dx.doi.org/10.1090/S0002994799024642
PII:
S 00029947(99)024642
Keywords:
Cantor sets,
difference sets,
thickness
Received by editor(s):
October 14, 1997
Published electronically:
September 20, 1999
Additional Notes:
Research supported in part by a grant from the Purdue Research Foundation
Article copyright:
© Copyright 1999
American Mathematical Society
