{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T21:28:31Z","timestamp":1776893311177,"version":"3.51.2"},"reference-count":22,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T00:00:00Z","timestamp":1759363200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Axioms"],"abstract":"<jats:p>The quasisymmetric minimality for fractal sets is a hot research topic for scholars focused on the fractal geometry and quasisymmetric mappings. In this paper, we study the quasisymmetric minimality on packing dimension for homogeneous perfect sets. By using some mathematical tools such as the mass distribution principle, we find that a special class of homogeneous perfect sets with packing dimension 1 is quasisymmetrically packing minimal. Our result generalizes the results in the references.<\/jats:p>","DOI":"10.3390\/axioms14100751","type":"journal-article","created":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T15:07:48Z","timestamp":1759417668000},"page":"751","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quasisymmetric Minimality on Packing Dimension for Homogeneous Perfect Sets"],"prefix":"10.3390","volume":"14","author":[{"given":"Shishuang","family":"Liu","sequence":"first","affiliation":[{"name":"College of Mathematics and Information Science, Guangxi University, Nanning 530004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanzhe","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Guangxi University, Nanning 530004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiaojiao","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Falconer, K. (1990). Fractal Geometry, Mathematical Foundations and Applications, John Wiley & Sons.","DOI":"10.2307\/2532125"},{"key":"ref_2","unstructured":"Wen, Z. (2000). Fractal Geometry-Mathematical Foundation, Shanghai Science and Technology Education Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3361","DOI":"10.1090\/S0002-9939-00-05433-2","article-title":"Sets of minimal Hausdorff dimension for quasiconformal maps","volume":"128","author":"Tyson","year":"2000","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1112\/jlms\/s2-6.3.504","article-title":"Hausdorff dimension and quasiconformal mappings","volume":"6","author":"Gehring","year":"1973","journal-title":"J. Lond. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/BF02392268","article-title":"The Lp-integrability of the partial derivatives of a quasiconformal mapping","volume":"130","author":"Gehring","year":"1973","journal-title":"Acta Math-Djursholm."},{"key":"ref_6","first-page":"397","article-title":"Quasiconformal mappings which increase dimension","volume":"24","author":"Bishop","year":"1999","journal-title":"Ann. Acad. Sci. Fenn. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1215\/S0012-7094-06-13411-7","article-title":"Conformal dimension does not assume values between zero and one","volume":"134","author":"Kovalev","year":"2006","journal-title":"Duke. Math. J."},{"key":"ref_8","first-page":"5","article-title":"Cantor sets minimal for quasisymmetric maps","volume":"41","author":"Hakobyan","year":"2006","journal-title":"J. Contemp. Math. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"139","DOI":"10.5186\/aasfm.2011.3608","article-title":"Quasisymmetrically minimal Moran sets and Hausdorff dimension","volume":"36","author":"Dai","year":"2011","journal-title":"Ann. Acad. Sci. Fenn. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1016\/j.topol.2014.10.005","article-title":"On quasisymmetric minimality of Cantor sets","volume":"178","author":"Wang","year":"2014","journal-title":"Topol. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"126783","DOI":"10.1016\/j.jmaa.2022.126783","article-title":"The continuity of dimensions and quasisymmetrical equivalence of parameterized homogeneous Moran sets","volume":"518","author":"Dai","year":"2023","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/j.jmaa.2013.04.085","article-title":"Quasisymmetric minimality on packing dimension for Moran sets","volume":"408","author":"Li","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1950101","DOI":"10.1142\/S0218348X19501019","article-title":"Quasisymmetric packing-minimality of Moran sets","volume":"27","author":"Li","year":"2019","journal-title":"Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2150043","DOI":"10.1142\/S0218348X21500432","article-title":"Quasisymmetrically minimal Moran sets on packing dimension","volume":"29","author":"Li","year":"2021","journal-title":"Fractals"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s10474-005-0175-1","article-title":"Hausdorff dimension of homogeneous perfect sets","volume":"107","author":"Wen","year":"2005","journal-title":"Acta Math. Hung."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/BF02896955","article-title":"Some dimensional results for homogeneous Moran sets","volume":"40","author":"Feng","year":"1997","journal-title":"Sci. China Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s10474-007-6145-z","article-title":"Packing dimensions of homogeneous perfect sets","volume":"118","author":"Wang","year":"2008","journal-title":"Acta Math. Hung."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Falconer, K. (1997). Techniques in Fractal Geometry, John Wiley & Sons.","DOI":"10.2307\/2533585"},{"key":"ref_19","first-page":"77","article-title":"Null sets for doubling and dyadic doubling measures","volume":"18","author":"Wu","year":"1993","journal-title":"Ann. Acad. Sci. Fenn. Ser. AI Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1849","DOI":"10.1007\/BF02901155","article-title":"Moran sets and Moran classes","volume":"46","author":"Wen","year":"2001","journal-title":"Chin. Sci. Bull."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2440003","DOI":"10.1142\/S0218348X24400036","article-title":"Mixed multifractal spectra of homogeneous Moran measures","volume":"32","author":"Hattab","year":"2024","journal-title":"Fractals"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"113818","DOI":"10.1016\/j.chaos.2023.113818","article-title":"On the multifractal measures and dimensions of image measures on a class of Moran sets","volume":"147","author":"Attia","year":"2023","journal-title":"Chaos Solitons Fractals"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/751\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T04:23:42Z","timestamp":1760070222000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/751"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,2]]},"references-count":22,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["axioms14100751"],"URL":"https:\/\/doi.org\/10.3390\/axioms14100751","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,2]]}}}