On Dynamics of Hyperbolic Rational Semigroups and Hausdorff Dimension of Julia sets

We consider about subsemigroups of $\mathrm{E}\mathrm{n}\mathrm{d}\overline{\mathbb{C}}$. We can define Julia sets, Fatou sets, etc. We show the backward sel&imilarity of Julia sets of finitely generated rational semigroups. If the post critical set of the semigroup is contained in a domain of Fatou set, the Julia set is a self similar set. Next, hyperbolic rational $\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{g}_{\mathrm{l}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}}$ have no wandering domains under a general assumption. If the hyperbolic rational semigroup is finitely gen-erated and satisfies some conditions, the limit functions of the semigroup on the Fatou set are only constant functions that take their values on post critical set. When the generators of a finitely generated hyperbolic rational $\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{g}_{1}\cdot \mathrm{o}\mathrm{u}\mathrm{p}$ are perturbed, the hyperbolicity is kept and the Jilia sets depend cotiniously on the generators. Further more, if the finitely generated $1\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{a}}1$ semigroup is hyperbolic and if the inverse images by the generators of the Julia set are mutually disjoint, then the Julia set moves by holomorphic motion

Editorial Board

Editors-in Chief Christopher J. Flann Jason P. Loble Articles Editors James M. Hughes Robert C. Lukes Jon O. Shields Citations Editors Jamie J. Gashwiler M. Scott Regan Managing Editor Sean S. Frampton Staff Natalie S. Adams Raymond J. Dearie, Jr. Sarah A. Dixon Lucas J. Foust Whitney L. Grubbs Kathleen S. Monzie William V. Roth Deirdre L. Runnette Justin W. Stark Julia M. Weddle Faculty Advisor Carl Tobia

Calanthe Officers

Shining Light Court of Calanthe No. 43, has elected the following officers: W. C.--Frances White. W. Ipx.--Bessie White. W. Iptr. Bessie Newsom. W. Iptr--Ross Turner. W. O.--Malinda Payne. S. D. Mary Ralls. J. D.--Jennie May. R. of D.--Bertha Payne. Reg. of A--Ella Newsom. Rec. of D. Elizabeth Copeland. W. Con.--Anna Turner. A. Con. Viola Booker. W. Escort--Leona Hamilton. W. Herald--Bessie Simmons. W. Protector--Julia Drake. Trustee, 18 months--Lon Whit

Principal Component Analysis of Molecular Clouds: Can CO reveal the dynamics?

We use Principal Component Analysis (PCA) to study the gas dynamics in numerical simulations of typical MCs. Our simulations account for the non-isothermal nature of the gas and include a simplified treatment of the time-dependent gas chemistry. We model the CO line emission in a post-processing step using a 3D radiative transfer code. We consider mean number densities n_0 = 30, 100, 300 cm^{-3} that span the range of values typical for MCs in the solar neighbourhood and investigate the slope \alpha_{PCA} of the pseudo structure function computed by PCA for several components: the total density, H2 density, 12CO density, 12CO J = 1 -> 0 intensity and 13CO J = 1 -> 0 intensity. We estimate power-law indices \alpha_{PCA} for different chemical species that range from 0.5 to 0.9, in good agreement with observations, and demonstrate that optical depth effects can influence the PCA. We show that when the PCA succeeds, the combination of chemical inhomogeneity and radiative transfer effects can influence the observed PCA slopes by as much as ~ +/- 0.1. The method can fail if the CO distribution is very intermittent, e.g. in low-density clouds where CO is confined to small fragments.Comment: 12 pages, 8 figures, accepted for publication in MNRA

Prime orbit theorems for expanding Thurston maps: Dirichlet series and orbifolds

We obtain an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumptions. More precisely, we show that the number of primitive periodic orbits, ordered by a weight on each point induced by an (eventually) positive real-valued H\"{o}lder continuous function on $S^2$ that is not cohomologous to a constant, is asymptotically the same as the well-known logarithmic integral. In particular, our results apply to postcritically-finite rational maps for which the Julia set is the whole Riemann sphere.Comment: 67 pages. This is the first of a series of 3 papers (together with arXiv:2312.06688 and arXiv:2312.06687), replacing arXiv:1804.0822

Ein Liederabend Recital Series, February 3, 1993

This is the concert program of the Ein Liederabend Recital Series performance on Wednesday, February 3, 1993 at 6:30 p.m., at the Concert Hall, 855 Commonwealth Avenue. Works performed were O bellissimi capelli by Andrea Falconiere, An die Nachtigall by Franz Schubert, "Nulla temer... Generoso chi sol brama" from "Scipione" by Georg Frideric Handel, A Nun Takes the Veil by Samuel Barber, The Crucifixion by S. Barber, Promiscuity by S. Barber, Tief im Herzen trag ich Pein by Robert Schumann, Sängers Trost by R. Schumann, Gretchen am Spinnrade by F. Schubert, Selections from "To Julia" by Roger Quilter, In der Fremde by R. Schumann, Chevaux de bois by Claude Debussy, and The Naughty Kitten by Modest Mussorgsky. Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund

Prime orbit theorems for expanding Thurston maps: Latt\`es maps and split Ruelle operators

We obtain an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumption. More precisely, we show that the number of primitive periodic orbits, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued H\"{o}lder continuous function on $S^2$ satisfying some additional regularity conditions, is asymptotically the same as the well-known logarithmic integral, with an exponential error term. In particular, our results apply to postcritically-finite rational maps for which the Julia set is the whole Riemann sphere. Moreover, a stronger result is obtained for Latt\`{e}s maps.Comment: 62 pages. This is the second of a series of 3 papers, replacing arXiv:1804.08221. arXiv admin note: substantial text overlap with arXiv:2312.0551