{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:29:44Z","timestamp":1776788984237,"version":"3.51.2"},"reference-count":42,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2016,3,4]],"date-time":"2016-03-04T00:00:00Z","timestamp":1457049600000},"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>The entropies of Shannon, R\u00e9nyi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance.<\/jats:p>","DOI":"10.3390\/e18030084","type":"journal-article","created":{"date-parts":[[2016,3,4]],"date-time":"2016-03-04T10:53:34Z","timestamp":1457088814000},"page":"84","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":192,"title":["Entropy and Fractal Antennas"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3320-1493","authenticated-orcid":false,"given":"Emanuel","family":"Guariglia","sequence":"first","affiliation":[{"name":"Department of Physics \u201cE. R. Caianiello\u201d, University of Salerno, Via Giovanni Paolo II, 84084 Fisciano, Italy"},{"name":"Division of Applied Mathematics, M\u00e4lardalen University, 721 23 V\u00e4ster\u00e5s, Sweden"}]}],"member":"1968","published-online":{"date-parts":[[2016,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Sys. Tech. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"487","DOI":"10.3390\/sym3030487","article-title":"Classifying entropy measures","volume":"3","author":"Garrido","year":"2011","journal-title":"Symmetry"},{"key":"ref_3","unstructured":"R\u00e9nyi, A. (July, January 20). On measures of information and entropy. Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, Berkeley, CA, USA."},{"key":"ref_4","first-page":"861","article-title":"A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces","volume":"119","author":"Kolmogorov","year":"1958","journal-title":"Dokl. Akad. Nauk SSSR (NS)"},{"key":"ref_5","unstructured":"Barnsley, M.F. (2012). Fractals Everywhere: New edition, Cambridge University Press."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Falconer, K.J. (2003). Fractal geometry: Mathematical Foundations and Applications, John Wiley & Sons.","DOI":"10.1002\/0470013850"},{"key":"ref_7","unstructured":"Balanis, C.A. (2005). Antenna Theory: Analysis and Design, John Wiley & Sons."},{"key":"ref_8","first-page":"1875","article-title":"Chaos in microwave antenna arrays","volume":"3","author":"Ram","year":"1996","journal-title":"IEEE MTT-S Int. Microw. Symp. Dig."},{"key":"ref_9","unstructured":"Valdivia, J.A. (1997). The Physics of High Altitude Lightning. [Ph.D. Thesis, The University of Maryland]."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1109\/MAP.2003.1232160","article-title":"A Discussion on the Significance of Geometry in Determining the Resonant Behavior of Fractal and Other Non-Euclidean Wire Antennas","volume":"45","author":"Best","year":"2003","journal-title":"IEEE Antennas Propag. Mag."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1109\/LAWP.2002.802584","article-title":"Operating Band Comparison of the Perturbated Sierpinski and Modified Parany Gasket Antennas","volume":"1","author":"Best","year":"2002","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Schuster, H.G., and Just, W. (2005). Deterministic Chaos: An Introduction, Wiley-VCH.","DOI":"10.1002\/3527604804"},{"key":"ref_13","unstructured":"Papoulis, A., and Pillai, S.U. (2002). Probability, Random Variables and Stochastic Processes, McGraw-Hill. Chapter 14."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"261","DOI":"10.3390\/e10030261","article-title":"Axiomatic characterization of information measures","volume":"10","year":"2008","journal-title":"Entropy"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1088\/0951-7715\/11\/4\/001","article-title":"Generalized entropies: R\u00e9nyi and correlation integral approach","volume":"11","author":"Takens","year":"1998","journal-title":"Nonlinearity"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.camwa.2013.01.017","article-title":"Entropy of fractal systems","volume":"66","author":"Zmenskal","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_17","unstructured":"Marsden, J.E., and Hoffman, M.J. (1993). Elementary Classical Analysis, W. H. Freeman and Company."},{"key":"ref_18","first-page":"3301","article-title":"On entropy, entropy-like quantities, and applications","volume":"20","author":"Keller","year":"2015","journal-title":"Discret. Contin. Dyn. Syst. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0378-4371(95)00294-4","article-title":"Generalized Kolmogorov entropy in the dynamics of the multifractal generation","volume":"223","author":"Zanette","year":"1996","journal-title":"Physica A"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Bhattacharya, R., and Majumdar, M. (2007). Random Dynamical Systems: Theory and Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511618628"},{"key":"ref_21","first-page":"55","article-title":"Lyapunov characteristic exponents and smooth ergodic theory","volume":"32","author":"Pesin","year":"1977","journal-title":"Uspeki Mat. Nauk"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3732","DOI":"10.3390\/e16073732","article-title":"On the connections of generalized entropies with Shannon and Kolmogorov-Sinai entropies","volume":"11","author":"Falniowski","year":"2014","journal-title":"Entropy"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1063\/1.1530990","article-title":"Regularities unseen, randomness observed: Levels of entropy convergence","volume":"13","author":"Crutchfield","year":"2003","journal-title":"Chaos"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF02773640","article-title":"Measure-theoretic complexity of ergodic systems","volume":"100","author":"Ferenczi","year":"1997","journal-title":"Isr. J. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1017\/S0143385700000341","article-title":"Topological complexity","volume":"20","author":"Blanchard","year":"2000","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.3934\/dcds.2003.9.1607","article-title":"Global and local complexity in weakly chaotic systems","volume":"9","author":"Galatolo","year":"2003","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1111\/j.1749-6632.1980.tb29710.x","article-title":"Power spectra and mixing properties of strange attractors","volume":"375","author":"Farmer","year":"1980","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Tricot, C. (1995). Curves and Fractal Dimension, Springer.","DOI":"10.1007\/978-1-4612-4170-6"},{"key":"ref_29","unstructured":"Mandelbrot, B.B. (1982). The Fractal Geometry of Nature, W. H. Freeman and Company."},{"key":"ref_30","unstructured":"Hata, M., Kigami, J., and Yamaguti, M. (1997). Mathematics of Fractals, American Mathematical Society."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Addison, P.S. (1997). Fractal and Chaos: An Illustrated Course, Institute of Physics Publishing.","DOI":"10.1201\/9780849384431"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1512\/iumj.1981.30.30055","article-title":"Fractals and self similarity","volume":"30","author":"Hutchinson","year":"1981","journal-title":"Indiana Univ. Math. J."},{"key":"ref_33","unstructured":"Cohen, N.L. Fractal\u2019s New Era in Military Antennas Design. Available online: http:\/\/defenseelectronicsmag.com\/site-files\/defenseelectronicsmag.com\/files\/archive\/rfdesign.com\/mag\/508RFDSF1.pdf."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1109\/LAWP.2007.902045","article-title":"A Modified Sierpinski Fractal Antenna for Multiband Application","volume":"6","author":"Hwang","year":"2007","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1109\/8.664115","article-title":"On the behavior of the Sierpinski multiband fractal antenna","volume":"46","author":"Romeu","year":"1998","journal-title":"IEEE Antennas Propag."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Peitgen, H., J\u00fcrgens, H., and Saupe, D. (2004). Chaos and Fractal: New Frontiers in Science, Springer.","DOI":"10.1007\/b97624"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1211","DOI":"10.1088\/0964-1726\/10\/6\/310","article-title":"Design of reconfigurable fractal antennas and RF-MEMS for spaced-based communication systems","volume":"10","author":"Vinoy","year":"2001","journal-title":"Smart Mater. Struct."},{"key":"ref_38","first-page":"3","article-title":"Fractal Geometry in Electromagnetics Applications\u2014From Antenna to Metamaterials","volume":"19","author":"Krzysztofik","year":"2013","journal-title":"Microw. Rev."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1142\/S0218348X99000098","article-title":"Self-Similarity and the Geometric Requirements for Frequency Independence in Antennae","volume":"7","author":"Hohlfeld","year":"1999","journal-title":"Fractals"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Sheluhin, O.I., Smolskiy, S.M., and Osin, A.V. (2007). Self-Similar Processes in Telecommunications, John Wiley & Sons.","DOI":"10.1002\/9780470062098"},{"key":"ref_41","unstructured":"Novak, M.N. (2004). Thinking in Patterns: Fractals and Related Phenomena in Nature, World Scientific Publishing."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1063\/1.166492","article-title":"Entropy computing via integration over fractal measures","volume":"10","year":"2000","journal-title":"Chaos"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/3\/84\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:20:11Z","timestamp":1760210411000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/3\/84"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,4]]},"references-count":42,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,3]]}},"alternative-id":["e18030084"],"URL":"https:\/\/doi.org\/10.3390\/e18030084","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,3,4]]}}}