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Geometry of self-similar measures

dc.contributor.authorMorán Cabré, Manuel
dc.contributor.authorRey Simo, José Manuel
dc.date.accessioned2023-06-21T01:35:29Z
dc.date.available2023-06-21T01:35:29Z
dc.date.issued1995-09
dc.description.abstractSelf-similar measures can be obtained by regarding the self similar set generated by a system of similitudes 1J.i = {<Pi}ieM as the probability space associated with an infinite process of Bernouilli trials with state space 1J.i. These measures are concentrated in Besicovitch sets, which are those sets composed oí points with given asymptotic frequencies in their generating similitudes. In this paper we obtain some geometric-size properties of self-similar measures. We generalize the expression of the Hausdorff and packing dimensiona of such measures to the case when M is countable. We give a precise answer to the problem of determining what packing measures are singular viith respect to self-slmilar measures. Both problems are solved by means of a technique which allows us to obtain efficient coverings of balls by cylinder sets. We also show that Besicovitch sets have infinite packing measure in their dimension.
dc.description.abstractLas medidas autosemejantes pueden obtenerse considerando el conjunto autosemejante generado por un sistema de semejanzas 1J.i = {<Pi}ieM, como el espacio de probabilidad natural asociado a un proceso infinito de ensayos de Bernouilli con espacio de estados 1J.i. Estas medidas están concentradas en los conjuntos de Besicovitch, que son los conjuntos de puntos cuyas secuencias de semejanzas generadoras tienen frecuencias asintóticas fijadas. En este artículo obtenemos algunas propiedades geométricas de las medidas autosemejantes. Por una parte, generalizamos la fórmula para las dimensiones Hausdorff y packing de las medidas autosemejantes al caso en que M es infinito numerable. También damos una clasificación muy precisa de las medidas packing que son singulares respecto a las medidas autosemejantes. Ambos problemas se resuelven mediante una técnica que permite recubrir de manera eficiente bolas mediante cilindros asociados a la construcción geométrica, Demostramos además que los conjuntos de Besicovitch tienen medida packing infinito en su dimensión.
dc.description.departmentDecanato
dc.description.facultyFac. de Ciencias Económicas y Empresariales
dc.description.refereedTRUE
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/26499
dc.identifier.relatedurlhttp://economicasyempresariales.ucm.es/working-papers-ccee
dc.identifier.urihttps://hdl.handle.net/20.500.14352/64094
dc.issue.number14
dc.language.isoeng
dc.page.total23
dc.publication.placeMadrid, España
dc.publisherFacultad de Ciencias Económicas y Empresariales. Decanato
dc.relation.ispartofseriesDocumentos de Trabajo de la Facultad de Ciencias Económicas y Empresariales
dc.rightsAtribución-NoComercial 3.0 España
dc.rights.accessRightsopen access
dc.rights.urihttps://creativecommons.org/licenses/by-nc/3.0/es/
dc.subject.keywordGeometry
dc.subject.keywordSelf-Similar Measures.
dc.subject.keywordGeometría
dc.subject.keywordMedidas autosemejantes.
dc.subject.ucmFunciones (Matemáticas)
dc.subject.unesco1202 Análisis y Análisis Funcional
dc.titleGeometry of self-similar measures
dc.typetechnical report
dc.volume.number1995
dcterms.referencesC. Bandt, Deterministic Fractals and Fractal Measures, Lecture Notes of the School on Measure Theory and Real Analysis, Grado, Italy, 1991. A.S. Besicovitch, On the Sum of Digits of Real Numbers Represented in the Dyadic System, Math. Annalen, 110 (1934), 321-30. P. BiIlingsley, Probability and Measure, Wiley, New York, 1978. G. Brown, G. Michon and J. Peyriere, On the Mulfifractal Analysis of Measures, J. Stat. Physics 66 (1992), 775-790. R. Cawley and R.D. Mauldin, Multifractal Decomposition of Moran Fractals, Adtl. in Math. 92 (1992), 196-236. C.D. Cutler, The HausdorffDimension Distribution ofFinite Measures in Euclidean Spaces, Canad. J. Math. (1986) 38 (6), 1459-1484. C.D. Cutler, Connecting Ergodicity and Dimension in Dynamical Systems, Ergodic Theory Dyn. Syst. (1990) 10 451-462. A. Deliu, J.S. Geronimo, R. Shonkwiler and D. Hardin, Dimensions Associated with Recurrent Self-similar Sets, Math. Proc. Camb. Phil. Soc. 110 (1991), 327-36. H.G. Egglestone, The Fractional Dimension of a Set Defined by Decimal Properties, Quart. J. of Math. Oxford Ser. 20 (1949), 31-6. K.J. Falconer, The Geometry 01 Fractal Sets, Cambridge University Press, 1985. H. Haase, Densities of Hausdorff Measures on Generalized Self-Similar Sets, preprint. J .E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-47. R.D. Mauldin and M. Urbanski, Dimensions and Measures in Infinnite Iterated Function Systems, Proc. London Math. Soc.(to appear). M. Morán, Hausdorff Measure of Infinitely Generated Self-Similar Sets, Monast. fur Math. (forthcoming). M. Morán and J.-M. Rey, Singularity of Self-Similar Measures with respect to Hausdorff Measures, preprint. L. Olsen, A Multifractal Formalism, Adv. in Math. (to appear). J.-M. Rey, Geometría de Medidas y Conjuntos Autosemejantes, Ph.D. thesis, Universidad Complutense de Madrid, 1995. C.A. Rogers and S.J. Taylor, Functions Continuous and Singular with respect to a Hausdorff Measure, Mathematika, 8 (1961), 1-31. A. Schief, Separation Properties for Self-Similar Seta, Proc. Amer. Math. Soc. 122 (1994), 111-115. D.W. Spear, Measures and Self-Similarity, Adv. in Math. 91(2) (1992), 143-157. S.J. Taylor, The Measure Theory of Random Fractals, Math. Proc. Camb. Phil. Soc. 100 (1986),383-406. S.J. Taylor and C. Tricot, Packing Measure, and its Evaluation for a Brownian Path, Trans. Amer. Math. Soc. 288(2) (1985), 679-699. C. Tricot, Two Definitions ofFractional Dimension, Math. Proc. Camb. Phil. Soc. 91 (1982), 57-74. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982. K.R. Wicks, Fractals and Hyperspaces, Lecture Notes in Math. 1492, Springer-Verlag, 1992. L.-S. Young, Dimension, Entropy and Liapunov Exponents, Ergodic Theory & Dynamical Systems 2 (1982), 109-24.
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relation.isAuthorOfPublication4749e40a-a59b-4407-8e28-6bc2340eee88
relation.isAuthorOfPublication.latestForDiscovery36e295dc-70b7-4ede-868c-a83357a04413

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