Horizons of Fractal Geometry and Complex Dimensions最新文献

An overview of complex fractal dimensions: from fractal strings to fractal drums, and back 复杂分形维的概述:从分形弦分形鼓，和回来

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 2018-03-28 DOI: 10.1090/conm/731/14677

M. Lapidus

{"title":"An overview of complex fractal dimensions:\u0000 from fractal strings to fractal drums, and\u0000 back","authors":"M. Lapidus","doi":"10.1090/conm/731/14677","DOIUrl":"https://doi.org/10.1090/conm/731/14677","url":null,"abstract":"Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with fractal boundary), in Sref{Sec:2}, and then in the higher-dimensional case of relative fractal drums and, in particular, of arbitrary bounded subsets of Euclidean space of $mathbb{R}^N$, for any integer $N geq 1$, in Sref{Sec:3}. \u0000Special attention is paid to discussing a variety of examples illustrating the general theory rather than to providing complete statements of the results and their proofs, for which we refer to the books cite{Lap-vF4} (2013, joint with M. van Frankenhuijsen) when $N=1$, and cite{LapRaZu1} (2017, joint with G. Radunovi' c and D. v{Z}ubrini'c) when $N geq 1$ is arbitrary. \u0000Finally, in an epilogue (Sref{Sec:4}), entitled \"From quantized number theory to fractal cohomology\", we briefly survey aspects of related work (motivated in part by the theory of complex fractal dimensions) of the author with H. Herichi (in the real case) cite{HerLap1}, along with cite{Lap8}, and with T. Cobler (in the complex case) cite{CobLap1}, respectively, as well as in the latter part of a book in preparation by the author, cite{Lap10}.","PeriodicalId":431279,"journal":{"name":"Horizons of Fractal Geometry and Complex\n Dimensions","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133716509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 10

Self-similar tilings of fractal blow-ups 分形爆破的自相似拼接

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 2017-09-27 DOI: 10.1090/CONM/731/14672

M. Barnsley, A. Vince

引用次数: 6

The Mass Transference Principle: Ten years on 传质原理:十年过去了

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 2017-04-21 DOI: 10.1090/CONM/731/14670

D. Allen, Sascha Troscheit

引用次数: 12

Regularly varying functions, generalized contents, and the spectrum of fractal strings 正则变函数，广义内容，分形弦谱

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 2017-03-27 DOI: 10.1090/CONM/731/14673

T. Eichinger, S. Winter

{"title":"Regularly varying functions, generalized\u0000 contents, and the spectrum of fractal\u0000 strings","authors":"T. Eichinger, S. Winter","doi":"10.1090/CONM/731/14673","DOIUrl":"https://doi.org/10.1090/CONM/731/14673","url":null,"abstract":"We revisit the problem of characterizing the eigenvalue distribution of the Dirichlet-Laplacian on bounded open sets $Omegasubsetmathbb{R}$ with fractal boundaries. It is well-known from the results of Lapidus and Pomerance cite{LapPo1} that the asymptotic second term of the eigenvalue counting function can be described in terms of the Minkowski content of the boundary of $Omega$ provided it exists. He and Lapidus cite{HeLap2} discussed a remarkable extension of this characterization to sets $Omega$ with boundaries that are not necessarily Minkowski measurable. They employed so-called generalized Minkowski contents given in terms of gauge functions more general than the usual power functions. The class of valid gauge functions in their theory is characterized by some technical conditions, the geometric meaning and necessity of which is not obvious. Therefore, it is not completely clear how general the approach is and which sets $Omega$ are covered. Here we revisit these results and put them in the context of regularly varying functions. Using Karamata theory, it is possible to get rid of most of the technical conditions and simplify the proofs given by He and Lapidus, revealing thus even more of the beauty of their results. Further simplifications arise from characterization results for Minkowski contents obtained in cite{RW13}. We hope our new point of view on these spectral problems will initiate some further investigations of this beautiful theory.","PeriodicalId":431279,"journal":{"name":"Horizons of Fractal Geometry and Complex\n Dimensions","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115103189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A measure-theoretic result for approximation by Delone sets 用Delone集逼近的一个测度理论结果

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 2017-02-16 DOI: 10.1090/conm/731/14671

M. Baake, A. Haynes

引用次数: 2

Eigenvalues of the Laplacian on domains with fractal boundary 分形边界域上拉普拉斯算子的特征值

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 1900-01-01 DOI: 10.1090/conm/731/14678

P. Pollack, C. Pomerance

引用次数: 0

The spectral operator and resonances 谱算子和共振

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 1900-01-01 DOI: 10.1090/CONM/731/14675

M. V. Frankenhuijsen

引用次数: 1

Dimensions of limit sets of Kleinian groups Kleinian群极限集的维数

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 1900-01-01 DOI: 10.1090/conm/731/14674

K. Falk

{"title":"Dimensions of limit sets of Kleinian\u0000 groups","authors":"K. Falk","doi":"10.1090/conm/731/14674","DOIUrl":"https://doi.org/10.1090/conm/731/14674","url":null,"abstract":"In this paper we give a brief survey of results on dimension gaps for limit sets of geometrically infinite Kleinian groups. We concentrate on an important notion from geometric group theory, amenability, as a criterion for the existence of such gaps. 1. Dimensions and invariants in conformal dynamics One goal often encountered in mathematics is to prove equality of independently defined invariants for large classes of a certain mathematical object. An instance of this in conformal dynamics has been the attempt to show that the critical exponent of the Poincaré series associated to a conformal dynamical system , e.g. a Kleinian group or a rational map on the Riemann sphere, coincides with the Hausdorff dimension of the corresponding limit set or Julia set, respectively. Since here we will be dealing mainly with Kleinian groups, i.e. discrete, torsion-free subgroups of the group of orientation preserving isometries of (n+ 1)-dimensional hyperbolic space H, n ∈ N, let us first explain briefly what these invariants are. The limit set L(G) of a Kleinian group G is the set of accumulation points of some and thus any orbit Gx of G, x ∈ H, and is a subset of the boundary ∂H of H due to the discreteness of G. The Poincaré series of G is defined as a Dirichlet series","PeriodicalId":431279,"journal":{"name":"Horizons of Fractal Geometry and Complex\n Dimensions","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117186116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Forward integrals and SDE with fractal noise 分形噪声下的正积分与SDE

Horizons of Fractal Geometry and Complex Dimensions Pub Date : 1900-01-01 DOI: 10.1090/CONM/731/14679

M. Zähle, E. Schneider

引用次数: 1