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Dissipative subshift5/8/2023 ![]() ![]() ![]() On the other hand, these techniques are limited to Hamiltonians with nearest-neighbor interaction and to one-dimensional systems or models that can be decomposed in one-dimensional systems. A short guideline through the workĪs discussed before, the methods of transfer matrices and trace maps are powerful tools. 42,48 While the condition for such an existence is already known as a “folklore theorem” (Theorem 2.8) among experts of symbolic dynamics, it seems that no explicit proof had been published so far. This topology is used more and more in symbolic dynamics. 7, it was shown that the Hausdorff topology on the set of orbit closures (or more generally dynamical subsystems) of a dynamical system was very useful in computing the spectra of associated Hamiltonians. The term approximation implies the use of a topology. In particular, a necessary and sufficient condition on the existence of periodic approximations is proved and used to build explicit algorithms permitting to compute them effectively. Building upon more general results, 7–9 this work investigates the consequences for symbolic dynamical systems, which provide a one-dimensional version of more general aperiodic media. This method has been commonly used even before any mathematically rigorous results were available to justify it. One strategy to get a hand on aperiodic systems consists in using periodic approximations. In light of this, the present paper gives a new result here that might help uncovering a solution. ![]() While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. Examples for the Fibonacci and the Golay–Rudin–Shapiro sequences are explicitly provided illustrating this discussion. In particular, nearest-neighbor correlation is not necessary. No restrictions on the structure of these operators other than general regularity assumptions are imposed. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson–Putnam complex, depending on the community. This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. ![]()
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