# Definition:Gauss Map

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## Definition

Let $\closedint 0 1$ denote the closed interval from $0$ to $1$.

The **Gauss map** $T : \closedint 0 1 \to \closedint 0 1$ is defined by:

- $\ds \map T x := \begin{cases} \fractpart {\dfrac 1 x} & : x \in \hointl 0 1 \\ 0 & : x = 0\end{cases}$

where $\fractpart \cdot$ denotes the fractional part.

This article, or a section of it, needs explaining.In particular: We have just changed the domain so that it includes rationals. Presumably there is a definition which does not include rationals. If we have multiple definitions, this must be handled so as to include both.Yes, I decided to take a more general definition from another book. But also in the first book, the map is often considered as adefined on $\closedint 0 1$, just neglecting the null set $\closedint 0 1 \cap \Q$. So, this definition is superior. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

## Also known as

It is also called **continued fractional map**.

## Source of Name

This entry was named for Carl Friedrich Gauss.

## Sources

- 2002: Michael Brin and Garrett Stuck:
*Introduction to Dynamical Systems*$1.6$ The Gauss Transformation

- 2011: Manfred Einsiedler and Thomas Ward:
*Ergodic Theory: with a view towards Number Theory*$3.2$ The Continued Fraction Map and the Gauss Measure

- Derwent, John and Weisstein, Eric W. "Gauss Map." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussMap.html