Editing Gauss–Kuzmin–Wirsing operator (section)

===Continuous spectrum===
The eigenvalues form a discrete spectrum, when the operator is limited to act on functions on the unit interval of the real number line.  More broadly, since the Gauss map is the shift operator on [[Baire space (set theory)|Baire space]] <math>\mathbb{N}^\omega</math>, the GKW operator can also be viewed as an operator on the function space <math>\mathbb{N}^\omega\to\mathbb{C}</math> (considered as a [[Banach space]], with basis functions taken to be the [[indicator function]]s on the [[cylinder set|cylinders]] of the [[product topology]]).  In the later case, it has a continuous spectrum, with eigenvalues in the unit disk <math>|\lambda|<1</math> of the complex plane.  That is, given the cylinder <math>C_n[b]= \{(a_1,a_2,\cdots) \in \mathbb{N}^\omega : a_n = b \}</math>, the operator G shifts it to the left: <math>GC_n[b] = C_{n-1}[b]</math>. Taking <math>r_{n,b}(x)</math> to be the indicator function which is 1 on the cylinder (when <math>x\in C_n[b]</math>), and zero otherwise, one has that <math>Gr_{n,b}=r_{n-1,b}</math>.  The series

:<math>f(x)=\sum_{n=1}^\infty \lambda^{n-1} r_{n,b}(x)</math>

then is an eigenfunction with eigenvalue <math>\lambda</math>. That is, one has <math>[Gf](x)=\lambda f(x)</math> whenever the summation converges: that is, when <math>|\lambda|<1</math>.

A special case arises when one wishes to consider the [[Haar measure]] of the shift operator, that is, a function that is invariant under shifts.  This is given by the [[Minkowski's question mark function|Minkowski measure]] <math>?^\prime</math>. That is, one has that <math>G?^\prime = ?^\prime</math>.<ref>{{cite arXiv |last1=Vepstas |first1=Linas |year=2008 |title=On the Minkowski Measure |eprint=0810.1265 |class=math.DS}}</ref>