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Thurston introduced emph{invariant (quadratic) laminations} in his 1984 preprint as a vehicle for understanding the connected Julia sets and the parameter space of quadratic polynomials. Important ingredients of his analysis of the angle doubling map $sigma_2$ on the unit circle $mathbb{S}^1$ were the Central Strip Lemma, non-existence of wandering polygons, the transitivity of the first return map on vertices of periodic polygons, and the non-crossing of minors of quadratic invariant laminations. We use Thurstons methods to prove similar results for emph{unicritical} laminations of arbitrary degree $d$ and to show that the set of so-called emph{minors} of unicritical laminations themselves form a emph{Unicritical Minor Lamination} $mathrm{UML}_d$. In the end we verify the emph{Fatou conjecture} for the unicritical laminations and extend the emph{Lavaurs algorithm} onto $mathrm{UML}_d$.
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We give a combinatorial criterion for a critical diameter to be compatible with a non-degenerate quadratic lamination.
Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of an invariant lamination by pullbacks of certain leaves, we describe how QML can be generated by properly understood pullbacks of certain minors. In particular, we show that the minors of all non-renormalizable quadratic laminations can be obtained by taking limits of pullbacks of minors from the main cardioid. This is the second, amended version of the paper, to appear in Contemporary Mathematics
We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(hat M, T^1mathcal{F})$ of a compact minimal lamination $(M,mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.
Let $K$ be the function field of a smooth, irreducible curve defined over $overline{mathbb{Q}}$. Let $fin K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r ge 1,$ is a power of the prime number $p$, and let $betain overline{K}$. For all $ninmathbb{N}cup{infty}$, the Galois groups $G_n(beta)=mathop{rm{Gal}}(K(f^{-n}(beta))/K(beta))$ embed into $[C_q]^n$, the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is not isotrivial, then $[[C_q]^infty:G_infty(beta)]<infty$ unless $beta$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$ and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields $bigcup_{n=1}^infty K(f_1^{-n}(beta))$ and $bigcup_{n=1}^infty K(f_2^{-n}(beta))$ are disjoint over a finite extension of $K$.
We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of the compactness theorem, we show that given a biholomorphism of $phi $ of a closed complex manifold $X$, some power $phi^k $ ($k>0$) fixes a measured Riemann surface lamination in $X$.
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