When Cantor sets intersect thickly
Authors:
Brian R. Hunt, Ittai Kan and James A. Yorke
Journal:
Trans. Amer. Math. Soc. 339 (1993), 869888
MSC:
Primary 28A80; Secondary 58F99
MathSciNet review:
1117219
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Abstract: The thickness of a Cantor set on the real line is a measurement of its "size". Thickness conditions have been used to guarantee that the intersection of two Cantor sets is nonempty. We present sharp conditions on the thicknesses of two Cantor sets which imply that their intersection contains a Cantor set of positive thickness.
 [1]
K.
J. Falconer, The geometry of fractal sets, Cambridge Tracts in
Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
(88d:28001)
 [2]
Roger
Kraft, Intersections of thick Cantor sets, Mem. Amer. Math.
Soc. 97 (1992), no. 468, vi+119. MR 1106988
(92i:28010)
 [3]
J.
M. Marstrand, Some fundamental geometrical properties of plane sets
of fractional dimensions, Proc. London Math. Soc. (3)
4 (1954), 257–302. MR 0063439
(16,121g)
 [4]
Pertti
Mattila, Hausdorff dimension and capacities of intersections of
sets in 𝑛space, Acta Math. 152 (1984),
no. 12, 77–105. MR 736213
(85g:28007), http://dx.doi.org/10.1007/BF02392192
 [5]
Sheldon
E. Newhouse, Nondensity of axiom 𝐴(𝑎) on
𝑆², Global Analysis (Proc. Sympos. Pure Math., Vol. XIV,
Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970,
pp. 191–202. MR 0277005
(43 #2742)
 [6]
Sheldon
E. Newhouse, Diffeomorphisms with infinitely many sinks,
Topology 13 (1974), 9–18. MR 0339291
(49 #4051)
 [7]
, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math. 50 (1979), 101151.
 [8]
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)
 [9]
Clark
Robinson, Bifurcation to infinitely many sinks, Comm. Math.
Phys. 90 (1983), no. 3, 433–459. MR 719300
(84k:58169)
 [10]
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
 [1]
 K. J. Falconer, The geometry of fractal sets, Cambridge Univ. Press, 1985. MR 867284 (88d:28001)
 [2]
 R. Kraft, Intersections of thick Cantor sets, Mem. Amer. Math. Soc., vol 97, no. 468, 1992. MR 1106988 (92i:28010)
 [3]
 J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3) 4 (1954), 257302. MR 0063439 (16:121g)
 [4]
 P. Mattila, Hausdorff dimension and capacities of intersections of sets in space, Acta Math. 152 (1984), 77105. MR 736213 (85g:28007)
 [5]
 S. Newhouse, Nondensity of axiom on , Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 191202. MR 0277005 (43:2742)
 [6]
 , Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 918. MR 0339291 (49:4051)
 [7]
 , The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math. 50 (1979), 101151.
 [8]
 , Lectures on dynamical systems, Progress in Math. 8, Birkhäuser, Boston, Mass., 1980, pp. 1114. MR 589590 (81m:58028)
 [9]
 C. Robinson, Bifurcation to infinitely many sinks, Comm. Math. Phys. 90 (1983), 433459. MR 719300 (84k:58169)
 [10]
 R. F. Williams, How big is the intersection of two thick Cantor sets?, Continuum Theory and Dynamical Systems (M. Brown, ed.), Proceedings of the 1989 Joint Summer Research Conference on Continua and Dynamics, Amer. Math. Soc., Providence, R.I., 1991. MR 1112813 (92f:58116)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199311172198
PII:
S 00029947(1993)11172198
Article copyright:
© Copyright 1993 American Mathematical Society
