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