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Let ${\mathbb D}:=\{(x_1,x_2)\in {\mathbb R}^2:x_1^2+x_2^2 < 1\}$ be the 2-dimensional unit disk and $\gamma \in H^{1/2} (\partial {\mathbb D},{\mathbb R}^3) \cap L^\infty (\partial {\mathbb D})$. For suitably small H>0, we consider the Dirichlet problem for H-systems \[ \hskip3.4cm \left\{\begin{array}{rlll} \Delta u &=& 2H u_{x_1} \wedge u_{x_2} &\quad {\rm in}\ {\mathbb D}\\[5pt] u &=& \gamma &\quad {\rm on}\ \partial {\mathbb D}\,.\hskip3.45cm {\rm (H)} \end{array} \right. \] It is well-known that (H) admits at least two distinct solutions for non-constant γ and small H>0. In this paper, we introduce conditions on γ and develop methods of finding at least three distinct solutions to (H) under such conditions on γ. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a few notions of inverse topological pressure $({\tilde P}^-, P^-, P_- )$, defined in terms of backward orbits (prehistories) instead of forward orbits. This inverse topological pressure has some properties similar to the regular (forward) pressure but, in general, if the map is not a homeomorphism, they do not coincide. In fact, there are several ways to define inverse topological pressure; for instance, we show that the Bowen type definition coincides with the one using spanning sets. Then we consider the case of a holomorphic map $f:{\mathbb P}^2 {\mathbb C}\to {\mathbb P}^2 {\mathbb C}$ which is Axiom A and such that its critical set does not intersect a particular basic set of saddle type Λ. We will prove that, under a technical condition, the Hausdorff dimension of the intersection between the local stable manifold and the basic set is equal to ts, i.e. $HD (W^s_\delta (x)\cap \Lambda)=t^s$, for all points x belonging to Λ. Here ts represents the unique zero of the function t→P-(tϕs), with P- denoting the inverse topological pressure and $\phi^s(y):=\log|Df|_{E_y^s}$, y∈Λ. In general, $HD (W^s_\delta (x)\cap\Lambda)$ will be estimated above by ts and below by $t^s_-$, where $t^s_-$ is the unique zero of the map t→P_(tϕs). As a corollary we obtain that, if the stable dimension is non-zero, then Λ must be a non-Jordan curve, and also, if f|Λ happens to be a homeomorphism (like in the examples from [13]), then the stable dimension cannot be zero. [ABSTRACT FROM AUTHOR]

In an earlier paper, the author introduced a collection of rings that control a categorification of the quantum sl(2) invariant of tangles. We prove that centers of these rings are isomorphics to the cohomology rings of the (n, n) Springer varieties and show that the braid group action in the derived category of modules over these rings descends to the Springer action of the symmetric group. [ABSTRACT FROM AUTHOR]