## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 35, Number 1 (2011), 67-77.

### Chain mixing endomorphisms are approximated by subshifts on the Cantor set

#### Abstract

Let *f* be a chain mixing continuous onto mapping from the Cantor set onto itself.
Let *g* be a homeomorphism on the Cantor set that is topologically conjugate to a
subshift. Then, homeomorphisms that are topologically conjugate to *g* approximate
*f* in the topology of uniform convergence if a trivial necessary condition on the
periodic points holds. In particular, if *f* is a chain mixing continuous onto
mapping from the Cantor set onto itself with a fixed point, then homeomorphisms on the
Cantor set that are topologically conjugate to a subshift approximate *f* in the
topology of uniform convergence. In addition, homeomorphisms on the Cantor set that are
topologically conjugate to a subshift without periodic points approximate any chain mixing
continuous onto mappings from the Cantor set onto itself. In particular, let *f* be a
homeomorphism on the Cantor set that is topologically conjugate to a full shift. Let
*g* be a homeomorphism on the Cantor set that is topologically conjugate to a
subshift. Then, a sequence of homeomorphisms that is topologically conjugate to *g*
approximates *f*.

#### Article information

**Source**

Tsukuba J. Math., Volume 35, Number 1 (2011), 67-77.

**Dates**

First available in Project Euclid: 19 July 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1311081449

**Digital Object Identifier**

doi:10.21099/tkbjm/1311081449

**Mathematical Reviews number (MathSciNet)**

MR2848816

**Zentralblatt MATH identifier**

1237.37015

**Subjects**

Primary: 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx] 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

**Keywords**

Cantor set shift subshift subshift of finite type chain mixing approximation

#### Citation

Shimomura, Takashi. Chain mixing endomorphisms are approximated by subshifts on the Cantor set. Tsukuba J. Math. 35 (2011), no. 1, 67--77. doi:10.21099/tkbjm/1311081449. https://projecteuclid.org/euclid.tkbjm/1311081449