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We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalization principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for a system with pure-point spectrum, the orbit separation dimension, and its relation to renormalization. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalization and Lyapunov exponents can be used to exclude the presence of absolutely continuous parts of the spectra.<\/jats:p>","DOI":"10.1107\/s2053273326003918","type":"journal-article","created":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T14:37:45Z","timestamp":1777559865000},"page":"294-304","update-policy":"https:\/\/doi.org\/10.1107\/cm_01","source":"Crossref","is-referenced-by-count":0,"title":["Renormalization techniques for inflation systems and some of their applications"],"prefix":"10.1107","volume":"82","author":[{"given":"Michael","family":"Baake","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02hpadn98","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Franz","family":"G\u00e4hler","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02hpadn98","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anna","family":"Klick","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02hpadn98","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Neil","family":"Ma\u00f1ibo","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02hpadn98","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jan","family":"Maz\u00e1\u010d","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02hpadn98","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"329","published-online":{"date-parts":[[2026,5,26]]},"reference":[{"key":"gau5007_bb1","doi-asserted-by":"crossref","unstructured":"Akiyama, S., Barge, M, Berth\u00e9, V., Lee, J.-Y. & Siegel, A. 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