Distortion of boundary sets under inner functions and applications

10 pages, no figures.-- MSC2000 codes: 30C85, 30D50.MR#: MR1183352 (93k:30014)Zbl#: Zbl 0765.30011An inner function is a bounded holomorphic function from the unit disc $\Delta$ of the complex plane such that the radial boundary values have modulus 1 a.e. . If $E$ is a Borel subset of $\partial\Delta$ we also define f(E)=\{e\sp{i\theta}/\lim\sb{r\to 1} f(re\sp{i\theta}) exists and belongs to $E\}$. Let M\sb \alpha, \text{cap}\sb \alpha and dim denote respectively the $\alpha$-dimensional content, $\alpha$- dimensional capacity and the Hausdorff dimension. In relation to the available results the authors in this paper prove that if $f$ is inner, $f(0)=0$, and $E$ is a Borel subset of $\partial\Delta$ then M\sb \alpha(f\sp{-1}(E)) \geq C\sb \alpha M\sb \alpha(E) and for $0\leq\alpha<1$, \text{cap}\sb \alpha(f\sp{-1}(E)) \geq C\sb \alpha \text{cap}\sb \alpha(E). An immediate consequence of course is \dim(f\sp{-1}(E))\geq \dim E. They also give examples to show that the inequalities cannot be reversed [source: Zentralblatt MATH].The first author was supported in part by a grant from CICYT, Ministerio de Educación y Ciencia, Spain.Publicad

Distortion of boundary sets under inner functions. II

33 pages, no figures.-- MSC2000 codes: 32A30, 30C85, 30D50.MR#: MR1379286 (97b:30035)Zbl#: Zbl 0847.32005We present a study of the metric transformation properties of inner functions of several complex variables. Along the way we obtain fractional dimensional ergodic properties of classical inner functions.Publicad

Trees, homology, and automorphism groups of RAAGs

We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this subgroup, based on the number and degree of a certain type of vertices, which we call deep. We then use combinatorial methods to analyze the average value of this Betti number, in terms of the size of the defining tree

Quantitative mixing results and inner functions

19 pages, no figures.-- MSC2000 codes: 30D05, 30D50, 37A05, 37A25, 37F10, 28D05, 11K55.MR#: MR2262783 (2007j:37003)Zbl#: Zbl 1125.30019We study in this paper estimates on the size of the sets of points which are well approximated by orbits of other points under certain dynamical systems. We apply the results obtained to the particular case of the dynamical system generated by inner functions in the unit disk of the complex plane.D. Pestana was supported by Grants BFM2003-04780 and BFM-2003-06335-C03-02, Ministerio de Ciencia y Tecnología, Spain. J. L. Fernández and M. V. Melián were supported by Grant BFM2003-04780 from Ministerio de Ciencia y Tecnología, Spain.Publicad

Fiscal harmonization in the presence of public inputs

Fiscal harmonization for the European Union member states is a goal that encounters major difficulties for its implementation. Each country faces a particular trade-off between fiscal revenues generated by taxation and the productive efficiency loss induced by their respective tax code. Countries for which a particular harmonized tax code requires more taxation will have to face an increased efficiency loss, whereas those required to decrease their taxes will have to face a loss in fiscal revenue. This paper provides a quantitative measure of these trade-offs, for a number of taxes and for the European Union member states, using a DGE model with public inputs. Calibration of the model for the EU-15 member states gives us the following results: i) The maximum tax revenue level is not far away from the current tax levels for most countries, ii) The cases of Sweden, Denmark and Finland are anomalous, as productive efficiency can be gained by lowering tax rates without affecting fiscal revenues, iii) In general, countries would obtain efficiency gains without changing fiscal revenues by reducing the capital tax and increasing the labor tax and iv) Capital tax harmonization to the average capital tax rate can be done with quite small changes in both fiscal revenues and output for the majority of countries.Fiscal harmonization, applied general equilibrium