#Singularity hidden map full

In the present paper, we provide a very simple argument that recovers thistheorem and thus gives an elementary solution to Problem 5 of.

#Singularity hidden map full

This remaining problem for zero measure of the spectrum of H О±,Оё was п¬Ѓnally solved by Avila-Krikorian, who employed a deep dynamical argument.We note that the argument of worked not for all О±, but for a full measure subset ofDiophantine О±. For the non-critical case,the proof for all О± of bounded type was given in, but the critical вЂњbounded-typeвЂќ caseremained diп¬ѓcult to crack.

The arguments of did notwork for this set, and even though the semi-classical analysis of Helп¬Ђer-SjВЁostrand appliedto some of this set for H О±,Оё, it did not apply to other such О±, including, most notably, thegolden mean - the subject of most numerical investigations. О±, again as Problem 5 in, which was to prove this conjecture for the remainingmeasure zero set of О±, namely, for О± of bounded type. Simon, п¬Ѓrst in his listof 15 problems in mathematical physics and then, after it was proved by Last for a.e. The Aubry-Andreconjecture stated that the measure of the spectrum of H О±,Оё,О» is equal to 4 | в€’ | О» ||, so to 0if О» = 1, for any irrational О±. Of course, it only makes sense to discuss upper bounds on the Hausdorп¬Ђ dimension ofa set on the real line once its Lebesgue measure is shown to be zero. We haveДЇor any irrational О± and real Оё, dim H ( Пѓ ( H О±,Оё )) в‰¤ /. Denote the spectrum of an operator K by Пѓ ( K ), the Lebesgue measure of a set A by | A |, and its Hausdorп¬Ђ dimension by dim H ( A ). This is also the п¬Ѓrst result on the fractal dimensionthat holds for more than a measure zero set of О±. We discuss the history of rigorous resultson the dimension in more detail below.The equality in the original conjecture can be viewed as two inequalities, and here weprove one of those for all irrational О±. Bellissard (e.g., ) that the dimension of the spectrum is a property that onlydepends on the tail in the continued fraction expansion of О± and thus should be the samefor a.e. There is also a conjecture attributedto J. However, all these are Lebesgue measure zero sets of О±, and the conjecture may still hold, in some sense. It has been rethought after rigorous and numerical studiesdemonstrated that the Hausdorп¬Ђ dimension can be less than 1/2 (and even be zero) for some О±, while packing/box counting dimension can be higher (even equal to one) forsome (in fact, of the same!) О±. A conjecture, some-times attributed to Thouless (e.g., ), and appearing already in the early 1980вЂ™s, is thatthe dimension is equal to 1/2. There have been many numerical and heuristic studies Department of Mathematics, UC Irvine, USA Department of Mathematics, Imperial College London, UK Also for any О» = 0.ДЎf its fractal dimension in physics literature (e.g., ). In our approach below, we use the fact that theproblem is two-dimensional in that we rely on an appropriate change of the gauge of themagnetic п¬Ѓeld.The spectrum of H О±,Оё for irrational О± is a Оё -independent fractal, beautifully depictedvia the Hofstadter butterп¬‚y.

The almost Mathieu operator H О±,Оё := H О±,Оё, at the critical value of the parameter О» is the one most important in physics, where it is also known as the Harper or Azbel-Hofstadter model (see e.g., and references therein), and also the one least understoodmathematically and even heuristically. The almost Mathieu family models anelectron on the two-dimensional square lattice in a uniform perpendicular magnetic п¬Ѓeld withп¬‚ux О± the critical case corresponds to the isotropic lattice, while О» = 1 corresponds toanisotropy. (1)with О» = 1, represents a transition from вЂњextended statesвЂќ for the subcritical case, 0 в‰¤ О». We prove almost Lipshitz continuity of spectra of singular quasiperiodic Jacobi matricesand obtain a representation of the critical almost Mathieu family that has a singularity.This allows us to prove that the Hausdorп¬Ђ dimension of its spectrum is not larger than1 / The critical almost Mathieu operator on в„“ ( Z ), that is( H О±,Оё,О» П† )( n ) = П† ( n в€’ДЎ) + П† ( n + 1) + 2 О» cos 2 ПЂ ( О±n + Оё ) П† ( n ), n =. Krasovsky In memory of David Thouless, 1934вЂ“2019 S e p Critical almost Mathieu operator: hidden singularity,gap continuity, and the Hausdorп¬Ђ dimension of thespectrum