Homoclinic points of mappings of the interval
Author:
Louis Block
Journal:
Proc. Amer. Math. Soc. 72 (1978), 576580
MSC:
Primary 58F20; Secondary 28D20, 54H20
MathSciNet review:
509258
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Abstract: Let f be a continuous map of a closed interval I into itself. A point is called a homoclinic point of f if there is a peridoic point p of f such that is in the unstable manifold of p, and p is in the orbit of x under , where n is the period of p. It is shown that f has a homoclinic point if and only if f has a periodic point whose period is not a power of 2. Furthermore, in this case, there is a subset X of I and a positive integer n, such that and there is a topological semiconjugacy of onto the full (onesided) shift on two symbols.
 [1]
Louis
Block, Continuous maps of the interval with
finite nonwandering set, Trans. Amer. Math.
Soc. 240 (1978),
221–230. MR 0474240
(57 #13887), http://dx.doi.org/10.1090/S00029947197804742402
 [2]
Louis
Block, Mappings of the interval with finitely
many periodic points have zero entropy, Proc.
Amer. Math. Soc. 67 (1977), no. 2, 357–360. MR 0467841
(57 #7692), http://dx.doi.org/10.1090/S00029939197704678413
 [3]
Rufus
Bowen and John
Franks, The periodic points of maps of the disk and the
interval, Topology 15 (1976), no. 4,
337–342. MR 0431282
(55 #4283)
 [4]
O.
M. Šarkovs′kiĭ, Coexistence of cycles of a
continuous mapping of the line into itself, Ukrain. Mat. Z.
16 (1964), 61–71 (Russian, with English summary). MR 0159905
(28 #3121)
 [5]
S.
Smale, Differentiable dynamical
systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 0228014
(37 #3598), http://dx.doi.org/10.1090/S000299041967117981
 [6]
P.
Štefan, A theorem of Šarkovskii on the existence of
periodic orbits of continuous endomorphisms of the real line, Comm.
Math. Phys. 54 (1977), no. 3, 237–248. MR 0445556
(56 #3894)
 [1]
 L. Block, Continuous maps of the interval with finite nonwandering set, Trans. Amer. Math. Soc. 240 (1978), 221230. MR 0474240 (57:13887)
 [2]
 , Mappings of the interval with finitely many periodic points have zero entropy, Proc. Amer. Math. Soc. 67 (1977), 357361. MR 0467841 (57:7692)
 [3]
 R. Bowen and J. Franks, The periodic points of maps of the disc and the interval, Topology 15 (1976), 337442. MR 0431282 (55:4283)
 [4]
 A. N. Sarkovskii, Coexistence of cycles of a continuous map of a line into itself, Ukrain. Mat. Ž. 16 (1964), 6171. MR 0159905 (28:3121)
 [5]
 S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747817. MR 0228014 (37:3598)
 [6]
 P. Stefan, A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. (to appear). MR 0445556 (56:3894)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919780509258X
PII:
S 00029939(1978)0509258X
Article copyright:
© Copyright 1978
American Mathematical Society
