Homoclinic points of mappings of the interval

Author:
Louis Block

Journal:
Proc. Amer. Math. Soc. **72** (1978), 576-580

MSC:
Primary 58F20; Secondary 28D20, 54H20

DOI:
https://doi.org/10.1090/S0002-9939-1978-0509258-X

MathSciNet review:
509258

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 (one-sided) shift on two symbols.

**[1]**L. Block,*Continuous maps of the interval with finite nonwandering set*, Trans. Amer. Math. Soc.**240**(1978), 221-230. MR**0474240 (57:13887)****[2]**-,*Mappings of the interval with finitely many periodic points have zero entropy*, Proc. Amer. Math. Soc.**67**(1977), 357-361. MR**0467841 (57:7692)****[3]**R. Bowen and J. Franks,*The periodic points of maps of the disc and the interval*, Topology**15**(1976), 337-442. 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), 61-71. MR**0159905 (28:3121)****[5]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747-817. 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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58F20,
28D20,
54H20

Retrieve articles in all journals with MSC: 58F20, 28D20, 54H20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0509258-X

Article copyright:
© Copyright 1978
American Mathematical Society