{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T13:25:03Z","timestamp":1780406703508,"version":"3.54.1"},"reference-count":36,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,11,26]],"date-time":"2019-11-26T00:00:00Z","timestamp":1574726400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive M\u00f6bius transforms of the line preserving unity (    q = 1     corresponds to the BG theory). Such transforms have the form     q \u21a6 ( a q + 1 \u2212 a ) \/ [ ( 1 + a ) q \u2212 a ]    , where a is a real number; the particular cases     a = \u2212 1     and     a = 0     yield, respectively,     q \u21a6 ( 2 \u2212 q )     and     q \u21a6 1 \/ q    , currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties.<\/jats:p>","DOI":"10.3390\/e21121155","type":"journal-article","created":{"date-parts":[[2019,11,26]],"date-time":"2019-11-26T10:57:27Z","timestamp":1574765847000},"page":"1155","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["M\u00f6bius Transforms, Cycles and q-triplets in Statistical Mechanics"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7681-7672","authenticated-orcid":false,"given":"Jean Pierre","family":"Gazeau","sequence":"first","affiliation":[{"name":"Astroparticles and Cosmology (UMR 7164), Sorbonne Paris Cit\u00e9, Univ Paris Diderot, 75205 Paris, France"},{"name":"Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9387-9194","authenticated-orcid":false,"given":"Constantino","family":"Tsallis","sequence":"additional","affiliation":[{"name":"Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Strasse 39, 1080 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gibbs, J.W. 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