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==Structure of the spectrum==
[[Image:Hofstadter's_butterfly.png|thumb|[[Hofstadter's butterfly]]]]
Another striking characteristic of the almost Mathieu operator is that its spectrum is a [[Cantor set]] for all irrational <math>\alpha</math> and <math>\lambda > 0</math>. This was shown by [[Artur Avila|Avila]] and [[Svetlana Jitomirskaya|Jitomirskaya]] solving the by-then famous 'Ten Martini Problem' <ref>{{Cite book|first1=A. |last1=Avila |first2=S. |last2=Jitomirskaya |title=The Ten Martini problem |volume=690 |pages=5–16 |year=2005 |arxiv=math/0503363 |bibcode=2006LNP...690....5A |doi=10.1007/3-540-34273-7_2 |chapter=Solving the Ten Martini Problem |series=Lecture Notes in Physics |isbn=978-3-540-31026-6 |s2cid=55259301 }}</ref> (also one of Simon's problems) after several earlier results (including generically<ref>{{cite journal |first1=J. |last1=Bellissard |first2=B. |last2=Simon |title=Cantor spectrum for the almost Mathieu equation |journal=[[Journal of Functional Analysis|J. Funct. Anal.]] |volume=48 |year=1982 |issue=3 |pages=408–419 |doi=10.1016/0022-1236(82)90094-5 |doi-access=free }}</ref> and almost surely<ref>{{cite journal |last=Puig |first=Joaquim |title=Cantor spectrum for the almost Mathieu operator |journal=Comm. Math. Phys. |volume=244 |year=2004 |issue=2 |pages=297–309 |doi=10.1007/s00220-003-0977-3 |arxiv=math-ph/0309004 |bibcode=2004CMaPh.244..297P |s2cid=120589515 }}</ref> with respect to the parameters).
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