## Nihonkai Mathematical Journal

- Nihonkai Math. J.
- Volume 25, Number 2 (2014), 119-125.

### Discontinuous maps whose iterations are continuous

#### Abstract

Let $X$ be a topological space and $f:X\to X$ a bijection. Let ${\mathcal C}(X,f)$ be a set of integers such that an integer $n$ is an element of ${\mathcal C}(X,f)$ if and only if the bijection $f^n:X\to X$ is continuous. A subset $S$ of the set of integers ${\mathbb Z}$ is said to be realizable if there is a topological space $X$ and a bijection $f:X\to X$ such that $S={\mathcal C}(X,f)$. A subset $S$ of ${\mathbb Z}$ containing $0$ is called a submonoid of ${\mathbb Z}$ if the sum of any two elements of $S$ is also an element of $S$. We show that a subset $S$ of ${\mathbb Z}$ is realizable if and only if $S$ is a submonoid of ${\mathbb Z}$. Then we generalize this result to any submonoid in any group.

#### Article information

**Source**

Nihonkai Math. J., Volume 25, Number 2 (2014), 119-125.

**Dates**

First available in Project Euclid: 26 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.nihmj/1427390302

**Mathematical Reviews number (MathSciNet)**

MR3326631

**Zentralblatt MATH identifier**

1318.54007

**Subjects**

Primary: 20M99: None of the above, but in this section 54C05: Continuous maps

**Keywords**

Continuous map discontinuous map bijection group monoid subgroup submonoid

#### Citation

Taniyama, Kouki. Discontinuous maps whose iterations are continuous. Nihonkai Math. J. 25 (2014), no. 2, 119--125. https://projecteuclid.org/euclid.nihmj/1427390302