Unfolding chaotic quadratic maps --- parameter dependence of natural measures

We consider perturbations of quadratic maps $f_a$ admitting an absolutely continuous invariant probability measure, where $a$ is in a certain positive measure set $\mathcal{A}$ of parameters, and show that in any neighborhood of any such an $f_a$, we find a rich fauna of dynamics. There are maps with periodic attractors as well as non-periodic maps whose critical orbit is absorbed by the continuation of any prescribed hyperbolic repeller of $f_a$. In particular, Misiurewicz maps are dense in $\mathcal{A}$. Almost all maps $f_a$ in the quadratic family is known to possess a unique natural measure, that is, an invariant probability measure $\mu_a$ describing the asymptotic distribution of almost all orbits. We discuss weak*-(dis)continuity properties of the map $a\mapsto \mu_a$ near the set $\mathcal{A}$, and prove that almost all maps in $\mathcal{A}$ have the property that $\mu_a$ can be approximated with measures supported on periodic attractors of certain nearby maps. On the other hand, for any $a \in \mathcal{A}$ and any periodic repeller $\Gamma_a$ of $f_a$, the singular measure supported on $\Gamma_a$ can also approximated with measures supported on nearby periodic attractors. It follows that $a\mapsto \mu_a$ is not weak*continuous on any full-measure subset of $(0,2]$. Some of these results extend to unimodal families with critical point of higher order, and even to not-too-flat flat topped families

Dynamics of Quadratic Families

Honors Project Paper, Department of Mathematics and Statistics, University of Minnesota Duluth, July 8, 2015This paper is based on the readings in the author's independent study on "advanced dynamical systems", and the author's mathematics honors project. It is a combination of the survey of some classical papers and the results from the research project. In the review part, none of the results are new and even less of them are due to the author; in the research part, we mainly focus the dynamics of the quadratic family along the real line. More specifically, in this paper we review and summarize the dynamics of one- and two- dimensional real quadratic maps from both topological and statistical viewpoints, and provide global pictures for their dynamics. Meanwhile, we briefly review the main results of the dynamics of one-dimensional complex quadratic maps under holomorphic singular perturbations, and provide recent research results about its dynamics under a nonholomorphic singular perturbation.Department of Mathematics and Statistics, University of Minnesota Dulut

Perturbations of Dirac operators

We study general conditions under which the computations of the index of a perturbed Dirac operator $D_{s}=D+sZ$ localize to the singular set of the bundle endomorphism $Z$ in the semi-classical limit $s\to \infty$. We show how to use Witten's method to compute the index of $D$ by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator $Z$. In particular, we provide examples of novel deformations of the de Rham operator to establish new results relating the Euler characteristic of a spin$^{c}$ manifold to maps between its even and odd spinor bundles. The paper contains a list of the current literature on the subject.Comment: 34 pages, improved results, new applications, literature list update

A surjection theorem for maps with singular perturbation and loss of derivatives

In this paper we introduce a new algorithm for solving perturbed nonlinear functional equations which admit a right-invertible linearization, but with an inverse that loses derivatives and may blow up when the perturbation parameter $\epsilon$ goes to zero. These equations are of the form $F\_\epsilon(u)=v$ with $F\_\epsilon(0)=0$, $v$ small and given, $u$ small and unknown. The main difference with the by now classical Nash-Moser algorithm is that, instead of using a regularized Newton scheme, we solve a sequence of Galerkin problems thanks to a topological argument. As a consequence, in our estimates there are no quadratic terms. For problems without perturbation parameter, our results require weaker regularity assumptions on $F$ and $v$ than earlier ones, such as those of Hormander. For singularly perturbed functionals, we allow $v$ to be larger than in previous works. To illustrate this, we apply our method to a nonlinear Schrodinger Cauchy problem with concentrated initial data studied by Texier-Zumbrun, and we show that our result improves significantly on theirs.Comment: Final version, to appear in Journal of the European Mathematical Society (JEMS

The Linear Information Coupling Problems

Many network information theory problems face the similar difficulty of single-letterization. We argue that this is due to the lack of a geometric structure on the space of probability distribution. In this paper, we develop such a structure by assuming that the distributions of interest are close to each other. Under this assumption, the K-L divergence is reduced to the squared Euclidean metric in an Euclidean space. In addition, we construct the notion of coordinate and inner product, which will facilitate solving communication problems. We will present the application of this approach to the point-to-point channel, general broadcast channel, and the multiple access channel (MAC) with the common source. It can be shown that with this approach, information theory problems, such as the single-letterization, can be reduced to some linear algebra problems. Moreover, we show that for the general broadcast channel, transmitting the common message to receivers can be formulated as the trade-off between linear systems. We also provide an example to visualize this trade-off in a geometric way. Finally, for the MAC with the common source, we observe a coherent combining gain due to the cooperation between transmitters, and this gain can be quantified by applying our technique.Comment: 27 pages, submitted to IEEE Transactions on Information Theor

Harmonic Splittings of Surfaces

We give a proof, using harmonic maps from disks to real trees, of Skora's theorem (Morgan-Otal (1993), Skora (1990), originally conjectured by Shalen): if G is the fundamental group of a surface of genus at least 2, then any small minimal G-action on a real tree is dual to the lift of a measured foliation. Analytic tools like the maximum principle are used to simplify the usual combinatorial topology arguments. Other analytic objects associated to a harmonic map, such as the Hopf differential and the moduli space of harmonic maps, are also introduced as tools for understanding the action of surface groups on trees.Comment: 28 page