{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:44:32Z","timestamp":1760233472666,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,6]],"date-time":"2021-01-06T00:00:00Z","timestamp":1609891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Centered polygonal lacunary functions are a type of lacunary function that exhibit behaviors that set them apart from other lacunary functions, this includes rotational symmetry. This work will build off of earlier studies to incorporate the automorphism group of the open unit disk D, which is a subgroup of the M\u00f6bius transformations. The behavior, dimension, dynamics, and sensitivity of filled-in Julia sets and Mandelbrot sets to variables will be discussed in detail. Additionally, several visualizations of this three-dimensional parameter space will be presented.<\/jats:p>","DOI":"10.3390\/sym13010091","type":"journal-article","created":{"date-parts":[[2021,1,6]],"date-time":"2021-01-06T20:45:42Z","timestamp":1609965942000},"page":"91","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Lacunary M\u00f6bius Fractals on the Unit Disk"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4661-9530","authenticated-orcid":false,"given":"L. K.","family":"Mork","sequence":"first","affiliation":[{"name":"Department of Mathematics, Concordia College, Moorhead, MN 56562, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Keith","family":"Sullivan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Vermont, Burlington, VT 05405, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8825-5424","authenticated-orcid":false,"given":"Darin J.","family":"Ulness","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Concordia College, Moorhead, MN 56562, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sullivan, K., Rutherford, D., and Ulness, D.J. (2019). Centered polygonal lacunary sequences. Mathematics, 7.","DOI":"10.3390\/math7100943"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"42","DOI":"10.3390\/fractalfract3030042","article-title":"Exploration of filled-in Julia sets arising from centered polygonal lacunary functions","volume":"3","author":"Mork","year":"2019","journal-title":"Fractal Fract."},{"key":"ref_3","unstructured":"Hille, E. (1959). Analytic Function Theory, Vol. I, Ginn and Company."},{"key":"ref_4","unstructured":"Hille, E. (1962). Analytic Function Theory, Vol. II, Ginn and Company."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Remmert, R. (1998). Classical Topics in Complex Function Theory, Springer.","DOI":"10.1007\/978-1-4757-2956-6"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"339","DOI":"10.4169\/math.mag.84.5.339","article-title":"Numbers simultaneously polygonal and centered polygonal","volume":"84","author":"Schlicker","year":"2011","journal-title":"Math. Mag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4545","DOI":"10.1021\/ic00220a025","article-title":"Magic numbers in polygonal clusters","volume":"24","author":"Teo","year":"1985","journal-title":"Inorg. Chem."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Deza, E., and Deza, M.-M. (2012). Figurate Numbers, World Scientific.","DOI":"10.1142\/9789814355490"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2050068","DOI":"10.1142\/S0218348X20500681","article-title":"Gaussian mixture noised random fractals with adversarial learning for automated creation of visual objects","volume":"28","author":"Xiang","year":"2020","journal-title":"Fractals"},{"key":"ref_10","first-page":"213","article-title":"A review on controlling of Julia and Mandelbrot sets","volume":"28","author":"Yadav","year":"2019","journal-title":"Int. J. Adv. Sci. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Abbas, M., Iqbal, H., and De la Sen, M. (2020). Generation of Julia and Mandelbrot Sets via Fixed Points. Symmetry, 12.","DOI":"10.3390\/sym12010086"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/BF02098448","article-title":"Similarity Between the Mandelbrot Set and Julia Sets","volume":"134","author":"Tan","year":"1990","journal-title":"Commun. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s13324-020-00357-4","article-title":"Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn","volume":"10","author":"Kawahira","year":"2020","journal-title":"Analysis Math. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"12167","DOI":"10.1109\/ACCESS.2019.2892013","article-title":"Mandelbrot and Julia Sets via Jungck-CR Iteration with s-Convexity","volume":"7","author":"Kwun","year":"2019","journal-title":"IEEE Access"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"35060","DOI":"10.1109\/ACCESS.2019.2904677","article-title":"Fractal generation in modified Jungck-S orbit","volume":"7","author":"Kwun","year":"2019","journal-title":"IEEE Access"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"76859","DOI":"10.1109\/ACCESS.2019.2920026","article-title":"Boundaries of filled Julia Sets in generalized Jungck Mann orbit","volume":"7","author":"Li","year":"2019","journal-title":"IEEE Access"},{"key":"ref_17","unstructured":"Feijs, L., and Toeters, M. (2019). Exploring a taxicab-based Mandelbrot-like set. Bridges 2019 Conference Proceedings, Johannes Kepler University."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Teoters, M., Feijs, L.M.G., Van Loenhout, D., Tieleman, C., Virtala, N., and Jaakson, G.K. (2019, January 11\u201313). Algorithmic fashion aesthetics: Mandelbrot. Proceedings of the ISWC \u201919: 23rd International Symposium on Wearable Computers, London, UK.","DOI":"10.1145\/3341163.3346946"},{"key":"ref_19","unstructured":"(2020). Mathematica 12, Wolfram Research."},{"key":"ref_20","unstructured":"Rainville, E.D. (1960). Special Functions, The MacMillen Company."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Peitgen, H.-O., J\u00fcrgens, H., and Saupe, D. (2004). Chaos and Fractals, Springer. [2nd ed.].","DOI":"10.1007\/b97624"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Falconer, K. (2004). Fractal Geometry: Mathematical Foundations and Applications, Wiley. [2nd ed.].","DOI":"10.1002\/0470013850"},{"key":"ref_23","unstructured":"Barnsley, M. (1993). Fractals Everywhere, Academic Press. [2nd ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"100019","DOI":"10.1016\/j.csfx.2019.100019","article-title":"The size of Mandelbrot bulbs","volume":"3","author":"Fowler","year":"2019","journal-title":"Chaos Solitons Fractals X"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/91\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:07:47Z","timestamp":1760159267000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/91"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,6]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010091"],"URL":"https:\/\/doi.org\/10.3390\/sym13010091","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,1,6]]}}}