[ { "id": "authors:7fhd0-f6071", "collection": "authors", "collection_id": "7fhd0-f6071", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230613-731854300.71", "type": "article", "title": "Etudes for the inverse spectral problem", "author": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Poltoratski", "given_name": "Alexei", "clpid": "Poltoratski-Alexei" } ], "abstract": "In this note, we study inverse spectral problems for canonical Hamiltonian systems, which encompass a broad class of second-order differential equations on a half-line. Our goal is to extend the classical results developed in the work of Marchenko, Gelfand\u2013Levitan, and Krein to broader classes of canonical systems and to illustrate the solution algorithms and formulae with a variety of examples. One of the main ingredients of our approach is the use of truncated Toeplitz operators, which complement the standard toolbox of the Krein\u2013de Branges theory of canonical systems.", "doi": "10.1112/jlms.12772", "issn": "0024-6107", "publisher": "London Mathematical Society", "publication": "Journal of the London Mathematical Society", "publication_date": "2023-06-27" }, { "id": "authors:6h7z1-n0b51", "collection": "authors", "collection_id": "6h7z1-n0b51", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211206-221748427", "type": "article", "title": "Bers slices in families of univalent maps", "author": [ { "family_name": "Lazebnik", "given_name": "Kirill", "orcid": "0000-0001-8963-4410", "clpid": "Lazebnik-Kirill" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Mukherjee", "given_name": "Sabyasachi", "orcid": "0000-0002-6868-6761", "clpid": "Mukherjee-Sabyasachi" } ], "abstract": "We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions \u03a3. This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial z \u21a6 ^(-d)z is the Schwarz reflection map arising from the corresponding map in \u03a3. We characterize the image of this embedding in \u03a3 as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic polynomial.", "doi": "10.1007/s00209-021-02871-y", "issn": "0025-5874", "publisher": "Springer", "publication": "Mathematische Zeitschrift", "publication_date": "2022-03", "series_number": "3", "volume": "300", "issue": "3", "pages": "2771-2808" }, { "id": "authors:w617a-5ag97", "collection": "authors", "collection_id": "w617a-5ag97", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210504-074618137", "type": "article", "title": "Schwarz reflections and anti-holomorphic correspondences", "author": [ { "family_name": "Lee", "given_name": "Seung-Yeop", "clpid": "Lee-Seung-Yeop" }, { "family_name": "Lyubich", "given_name": "Mikhail", "clpid": "Lyubich-Mikhail" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Mukherjee", "given_name": "Sabyasachi", "clpid": "Mukherjee-Sabyasachi" } ], "abstract": "In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in [14], [15]. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of this straightening highlighting the issues related to its anti-holomorphic nature. We complete the discussion by comparing our family with the classical Bullett-Penrose family of matings between groups and rational maps induced by holomorphic correspondences. More precisely, we show that the Schwarz reflections give rise to anti-holomorphic correspondences that are matings of parabolic anti-rational maps with the abstract modular group. We further illustrate our mating framework by studying the correspondence associated with the Schwarz reflection map of a deltoid.", "doi": "10.1016/j.aim.2021.107766", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "2021-07-16", "volume": "385", "pages": "Art. No. 107766" }, { "id": "authors:6rr9h-36280", "collection": "authors", "collection_id": "6rr9h-36280", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210715-154348375", "type": "article", "title": "Univalent polynomials and Hubbard trees", "author": [ { "family_name": "Lazebnik", "given_name": "Kirill", "orcid": "0000-0001-8963-4410", "clpid": "Lazebnik-Kirill" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Mukherjee", "given_name": "Sabyasachi", "orcid": "0000-0002-6868-6761", "clpid": "Mukherjee-Sabyasachi" } ], "abstract": "We study rational functions f of degree d+1 such that f is univalent in the exterior unit disc, and the image of the unit circle under f has the maximal number of cusps (d+1) and double points (d\u22122). We introduce a bi-angled tree associated to any such f. It is proven that any bi-angled tree is realizable by such an f, and moreover, f is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such f are in natural 1:1 correspondence with anti-holomorphic polynomials of degree d with d\u22121 distinct, fixed critical points (classified by their Hubbard trees).", "doi": "10.1090/tran/8387", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "2021-07", "series_number": "7", "volume": "374", "issue": "7", "pages": "4839-4893" }, { "id": "authors:1g0qn-4qw13", "collection": "authors", "collection_id": "1g0qn-4qw13", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191011-112435530", "type": "article", "title": "Scaling limits of random normal matrix processes at singular boundary points", "author": [ { "family_name": "Ameur", "given_name": "Yacin", "clpid": "Ameur-Yacin" }, { "family_name": "Kang", "given_name": "Nam-Gyu", "clpid": "Kang-Nam-Gyu" }, { "family_name": "Makarov", "given_name": "Nikolai", "clpid": "Makarov-N-G" }, { "family_name": "Wennman", "given_name": "Aron", "clpid": "Wennman-A" } ], "abstract": "We introduce a method for taking microscopic limits of normal matrix ensembles and apply it to study the behaviour near certain types of singular points on the boundary of the droplet. Our investigation includes ensembles without restrictions near the boundary, as well as hard edge ensembles, where the eigenvalues are confined to the droplet. We establish in both cases existence of new types of determinantal point fields, which differ from those which can appear at a regular boundary point, or in the bulk.", "doi": "10.1016/j.jfa.2019.108340", "issn": "0022-1236", "publisher": "Elsevier", "publication": "Journal of Functional Analysis", "publication_date": "2020-02-01", "series_number": "3", "volume": "278", "issue": "3", "pages": "Art. No. 108340" }, { "id": "authors:t28wy-tnq87", "collection": "authors", "collection_id": "t28wy-tnq87", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191205-100207235", "type": "article", "title": "Two-spectra theorem with uncertainty", "author": [ { "family_name": "Makarov", "given_name": "Nikolai", "clpid": "Makarov-N-G" }, { "family_name": "Poltoratski", "given_name": "Alexei", "clpid": "Poltoratski-A" } ], "abstract": "The goal of this paper is to combine ideas from the theory of mixed spectral problems for differential operators with new results in the area of the Uncertainty Principle in Harmonic Analysis (UP). Using recent solutions of Gap and Type Problems of UP we prove a version of Borg's two-spectra theorem for Schr\u00f6dinger operators, allowing uncertainty in the placement of the eigenvalues. We give a formula for the exact \"size of uncertainty,\" calculated from the lengths of the intervals where the eigenvalues may occur. Among other applications, we describe pairs of indeterminate operators in the three-interval case of the mixed spectral problem. At the end of the paper we discuss further questions and open problems.", "doi": "10.4171/JST/276", "issn": "1664-039X", "publisher": "European Mathematical Society", "publication": "Journal of Spectral Theory", "publication_date": "2019-09-05", "series_number": "4", "volume": "9", "issue": "4", "pages": "1249-1285" }, { "id": "authors:jecbx-ee723", "collection": "authors", "collection_id": "jecbx-ee723", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180402-110529752", "type": "article", "title": "Rescaling Ward Identities in the Random Normal Matrix Model", "author": [ { "family_name": "Ameur", "given_name": "Yacin", "clpid": "Ameur-Yacin" }, { "family_name": "Kang", "given_name": "Nam-Gyu", "clpid": "Kang-Nam-Gyu" }, { "family_name": "Makarov", "given_name": "Nikolai", "clpid": "Makarov-N-G" } ], "abstract": "We study spacing distribution for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward's (or the \"rescaled loop\") equation\u2014an identity satisfied by all sequential limits of the rescaled one-point functions.", "doi": "10.1007/s00365-018-9423-9", "issn": "0176-4276", "publisher": "Springer", "publication": "Constructive Approximation", "publication_date": "2019-08", "series_number": "1", "volume": "50", "issue": "1", "pages": "63-127" }, { "id": "authors:vt8t3-7f142", "collection": "authors", "collection_id": "vt8t3-7f142", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160428-075058722", "type": "article", "title": "Topology of quadrature domains", "author": [ { "family_name": "Lee", "given_name": "Seung-Yeop", "clpid": "Lee-Seung-Yeop" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "abstract": "We address the problem of topology of quadrature domains, namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity.", "doi": "10.1090/jams828", "issn": "0894-0347", "publisher": "American Mathematical Society", "publication": "Journal of the American Mathematical Society", "publication_date": "2016-05-11", "series_number": "2", "volume": "29", "issue": "2", "pages": "333-369" }, { "id": "authors:8kdqz-6kf82", "collection": "authors", "collection_id": "8kdqz-6kf82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150619-092050374", "type": "article", "title": "Random normal matrices and Ward identities", "author": [ { "family_name": "Ameur", "given_name": "Yacin", "clpid": "Ameur-Yacin" }, { "family_name": "Hedenmalm", "given_name": "Haakan", "clpid": "Hedenmalm-Haakan" }, { "family_name": "Makarov", "given_name": "Nikolai", "clpid": "Makarov-N-G" } ], "abstract": "We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.", "doi": "10.1214/13-AOP885", "issn": "0091-1798", "publisher": "Institute of Mathematical Statistics", "publication": "Annals of Probability", "publication_date": "2015-05", "series_number": "3", "volume": "43", "issue": "3", "pages": "1157-1201" }, { "id": "authors:0cgjx-qc996", "collection": "authors", "collection_id": "0cgjx-qc996", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130625-102609096", "type": "article", "title": "Coulomb gas ensembles and Laplacian growth", "author": [ { "family_name": "Hedenmalm", "given_name": "H\u00e5kan", "clpid": "Hedenmalm-Haakan" }, { "family_name": "Makarov", "given_name": "Nikolai", "clpid": "Makarov-N-G" } ], "abstract": "We consider weight functions Q : \u2102\u2192\u211d that are locally in a suitable Sobolev space and impose a logarithmic growth condition from below. We use Q as a confining potential in the model of one-component plasma (2-dimensional Coulomb gas) and study the configuration of the electron cloud as the number n of electrons tends to infinity, while the confining potential is rescaled: we use mQ in place of Q and let m tend to infinity as well. We show that if m and n tend to infinity in a proportional fashion, with n/m\u2192t, where 0", "doi": "10.1112/plms/pds032", "issn": "0024-6115", "publisher": "London Mathematical Society", "publication": "Proceedings of the London Mathematical Society", "publication_date": "2013-04", "series_number": "4", "volume": "106", "issue": "4", "pages": "859-907" }, { "id": "authors:w6yqv-pq051", "collection": "authors", "collection_id": "w6yqv-pq051", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20131119-095918059", "type": "article", "title": "Gaussian free field and conformal field theory", "author": [ { "family_name": "Kang", "given_name": "Nam-Gyu", "clpid": "Kang-Nam-Gyu" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "abstract": "In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie derivatives. Based on this approach, we explain some equations of conformal field theory and outline their relation to SLE theory.", "doi": "10.48550/arXiv.1101.1024", "issn": "0303-1179", "publisher": "Societe Mathematique de France", "publication": "Ast\u00e9risque", "publication_date": "2013", "volume": "353", "pages": "1-136" }, { "id": "authors:07p76-kyj28", "collection": "authors", "collection_id": "07p76-kyj28", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110808-135723507", "type": "article", "title": "Fluctuations of eigenvalues of random normal matrices", "author": [ { "family_name": "Ameur", "given_name": "Yacin", "clpid": "Ameur-Yacin" }, { "family_name": "Hedenmalm", "given_name": "H\u00e5kan", "clpid": "Hedenmalm-Haakan" }, { "family_name": "Makarov", "given_name": "Nikolai", "clpid": "Makarov-N-G" } ], "abstract": "In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane\u2014the \"droplet.\" We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.", "doi": "10.1215/00127094-1384782", "issn": "0012-7094", "publisher": "Duke University Press", "publication": "Duke Mathematical Journal", "publication_date": "2011-07-15", "series_number": "1", "volume": "159", "issue": "1", "pages": "31-81" }, { "id": "authors:hr6kv-j6r42", "collection": "authors", "collection_id": "hr6kv-j6r42", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20101111-144213512", "type": "article", "title": "Berezin Transform in Polynomial Bergman Spaces", "author": [ { "family_name": "Ameur", "given_name": "Yacin", "clpid": "Ameur-Yacin" }, { "family_name": "Hedenmalm", "given_name": "H\u00e5kan", "clpid": "Hedenmalm-Haakan" }, { "family_name": "Makarov", "given_name": "Nikolai", "clpid": "Makarov-N-G" } ], "abstract": "Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let K_(m,n) denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L^2-norm with respect to the measure e^(-mQ) dA. Here dA is normalized area measure, and m is a positive real scaling parameter. The (polynomial) Berezin measure dB^()_(m,n)(z)= K_(m,n)(z_0,z_0)^(-1) \u2502K_(m,n)(z,z_0)\u2502^2e^(-mQ(z)) dA(z) for the point z_0 is a probability measure that defines the (polynomial) Berezin transform B_(m,n f)(z_0)= \u0283C f dB^()_(m,n) for continuous f \u0454 L^\u221e(C). We analyze the semiclassical limit of the Berezin measure (and transform) as m \u2192 +\u221e while n = m \u03c4 + o(1), where \u03c4 is fixed, positive, and real. We find that the Berezin measure for z_0 converges weak-star to the unit point mass at the point z_0 provided that \u0394Q(z_0) > 0 and that z_0 is contained in the interior of a compact set S_ \u03c4, defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane. For points z_0 \u0454 C\\S _\u03c4 , the Berezin measure cannot converge to the point mass at z_0. In the model case Q(z)= \u2502z\u2502^2, when S_ \u03c4 is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z_0 relative to C\\S_ \u03c4. Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L^2 -estimates for the equation \u2202[overscore]u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at \u221e.", "doi": "10.1002/cpa.20329", "issn": "0010-3640", "publisher": "Wiley", "publication": "Communications on Pure and Applied Mathematics", "publication_date": "2010-12", "series_number": "12", "volume": "63", "issue": "12", "pages": "1533-1584" }, { "id": "authors:17220-b6160", "collection": "authors", "collection_id": "17220-b6160", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100611-112618541", "type": "article", "title": "Beurling-Malliavin theory for Toeplitz kernels", "author": [ { "family_name": "Makarov", "given_name": "N.", "clpid": "Makarov-N-G" }, { "family_name": "Poltoratski", "given_name": "A.", "clpid": "Poltoratski-A" } ], "abstract": "We consider the family of Toeplitz operators T_(JS[overbar]^a) acting in the Hardy space H^2 in the upper halfplane; J and S are given meromorphic inner functions, and a is a real parameter. In the case where the argument of S has a power law type behavior on the real line, we compute the critical value c(J, S) = inf{a: ker T_(JS[overbar]^a) \u2260 0}\nThe formula for c(J,S) generalizes the Beurling-Malliavin theorem on the radius of completeness for a system of exponentials.", "doi": "10.1007/s00222-010-0234-2", "issn": "0020-9910", "publisher": "Springer", "publication": "Inventiones Mathematicae", "publication_date": "2010-06", "series_number": "3", "volume": "180", "issue": "3", "pages": "443-480" }, { "id": "authors:9hvkj-w3112", "collection": "authors", "collection_id": "9hvkj-w3112", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100810-141434585", "type": "book_section", "title": "Off-critical lattice models and massive SLEs", "book_title": "XVIth International Congress on Mathematical Physics", "author": [ { "family_name": "Makarov", "given_name": "Nikolai", "clpid": "Makarov-N-G" }, { "family_name": "Smirnov", "given_name": "Stanislav", "clpid": "Smirnov-Stanislav" } ], "abstract": "We suggest how versions of Schramm's SLE can be used to describe the scaling limit of\nsome off-critical 2D lattice models. Many open questions remain.", "doi": "10.48550/arXiv.0909.5377", "isbn": "978-981-4304-62-7", "publisher": "World Scientific", "place_of_publication": "Hackensack, NJ", "publication_date": "2010", "pages": "362-371" }, { "id": "authors:8jnjw-jey06", "collection": "authors", "collection_id": "8jnjw-jey06", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110819-095049741", "type": "book_section", "title": "Meromorphic Inner Functions, Toeplitz Kernels and the Uncertainty Principle", "book_title": "Perspectives in Analysis: Essays in Honor of Lennart Carleson's 75th Birthday", "author": [ { "family_name": "Makarov", "given_name": "N.", "clpid": "Makarov-N-G" }, { "family_name": "Poltoratski", "given_name": "A.", "clpid": "Poltoratski-Alexei" } ], "contributor": [ { "family_name": "Benedicks", "given_name": "M.", "clpid": "Benedicks-M" }, { "family_name": "Jones", "given_name": "P. W.", "clpid": "Jones-P-W" }, { "family_name": "Smirnov", "given_name": "Stanislav", "clpid": "Smirnov-Stanislav" } ], "abstract": "This paper touches upon several traditional topics of 1D linear complex analysis\nincluding distribution of zeros of entire functions, completeness problem for\ncomplex exponentials and for other families of special functions, some problems\nof spectral theory of selfadjoint differential operators. Their common\nfeature is the close relation to the theory of complex Fourier transform of\ncompactly supported measures or, more generally, Fourier\u2013Weyl\u2013Titchmarsh\ntransforms associated with selfadjoint differential operators with compact resolvent.", "doi": "10.1007/3-540-30434-7_10", "isbn": "3-540-30432-0", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "2005", "pages": "185-252" }, { "id": "authors:mpxk3-e7387", "collection": "authors", "collection_id": "mpxk3-e7387", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111019-084131204", "type": "article", "title": "On Thermodynamics of Rational Maps. II: Non-Recurrent Maps", "author": [ { "family_name": "Makarov", "given_name": "N.", "clpid": "Makarov-N-G" }, { "family_name": "Smirnov", "given_name": "S.", "clpid": "Smirnov-Stanislav" } ], "abstract": "The pressure function p(t) of a non-recurrent map is real analytic on some interval (0,t_*) with t_* strictly greater than the dimension of the Julia set. The proof is an adaptation of the well known tower techniques to the complex dynamics situation. In general, p(t) need not be analytic on the whole positive axis.", "doi": "10.1112/S0024610702003964", "issn": "0024-6107", "publisher": "London Mathematical Society", "publication": "Journal of the London Mathematical Society", "publication_date": "2003-04", "volume": "67", "pages": "417-432" }, { "id": "authors:gna1q-8c681", "collection": "authors", "collection_id": "gna1q-8c681", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200323-132322336", "type": "article", "title": "Laplacian path models", "author": [ { "family_name": "Carleson", "given_name": "L.", "clpid": "Carleson-L" }, { "family_name": "Makarov", "given_name": "N.", "clpid": "Makarov-N-G" } ], "abstract": "In this paper, we study two growth models in the complex plane--the needle and the geodesic \u03b7-models, defined below.", "doi": "10.1007/bf02868471", "issn": "0021-7670", "publisher": "Hebrew University Magnes Press", "publication": "Journal d'Analyse Math\u00e9matique", "publication_date": "2002-12", "series_number": "1", "volume": "87", "issue": "1", "pages": "103-150" }, { "id": "authors:jap39-n8364", "collection": "authors", "collection_id": "jap39-n8364", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200127-141129762", "type": "article", "title": "On \"Thermodynamics\" of Rational Maps I. Negative Spectrum", "author": [ { "family_name": "Makarov", "given_name": "N.", "clpid": "Makarov-N-G" }, { "family_name": "Smirnov", "given_name": "S.", "clpid": "Smirnov-Stanislav" } ], "abstract": "We study the pressure spectrum P(t) of the maximal measure for arbitrary rational maps. We also consider its modified version which is defined by means of the variational principle with respect to non-atomic invariant measures. It is shown that for negative values of t, the modified spectrum has all major features of the hyperbolic case (analyticity, the existence of a spectral gap for the corresponding transfer operator, rigidity properties, etc). The spectrum P(t) can be computed in terms of . Their Legendre transforms are the Hausdorff and the box-counting dimension spectra of the maximal measure respectively. This work is closely related to a paper [32] by D. Ruelle.", "doi": "10.1007/s002200050833", "issn": "0010-3616", "publisher": "Springer", "publication": "Communications in Mathematical Physics", "publication_date": "2000-05", "series_number": "3", "volume": "211", "issue": "3", "pages": "705-743" }, { "id": "authors:yxqjj-rtf43", "collection": "authors", "collection_id": "yxqjj-rtf43", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180413-112754476", "type": "article", "title": "Singular continuous spectrum is generic", "author": [ { "family_name": "Del Rio", "given_name": "R.", "clpid": "Del Rio-R" }, { "family_name": "Jitomirskaya", "given_name": "S.", "clpid": "Jitomirskaya-S" }, { "family_name": "Makarov", "given_name": "N.", "clpid": "Makarov-N-G" }, { "family_name": "Simon", "given_name": "B.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" } ], "abstract": "In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense G_\u03b4.", "doi": "10.1090/S0273-0979-1994-00518-X", "issn": "0273-0979", "publisher": "American Mathematical Society", "publication": "Bulletin of the American Mathematical Society", "publication_date": "1994-10", "series_number": "2", "volume": "31", "issue": "2", "pages": "208-212" }, { "id": "authors:vjsg7-gmx86", "collection": "authors", "collection_id": "vjsg7-gmx86", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20171120-161021287", "type": "article", "title": "Operators with singular continuous spectrum: II. Rank one operators", "author": [ { "family_name": "Del Rio", "given_name": "R.", "clpid": "Del Rio-R" }, { "family_name": "Makarov", "given_name": "N.", "clpid": "Makarov-N-G" }, { "family_name": "Simon", "given_name": "B.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" } ], "abstract": "For an operator, A, with cyclic vector \u03d5, we study A+\u03bbP, where P is the rank one projection onto multiples of \u03d5. If [\u03b1,\u03b2] \u2282 spec (A) and A has no a.c. spectrum, we prove that A+\u03bbP has purely singular continuous spectrum on (\u03b1,\u03b2) for a dense G_\u03b4 of \u03bb's.", "doi": "10.1007/BF02099737", "issn": "0010-3616", "publisher": "Springer", "publication": "Communications in Mathematical Physics", "publication_date": "1994-10", "series_number": "1", "volume": "165", "issue": "1", "pages": "59-67" } ]