Locally compact flows on connected manifolds

In this paper, we completely characterize locally compact flows G G of homeomorphisms of connected manifolds M M by proving that they are either circle groups or real groups. For M = R m M = \mathbb R^m , we prove that every recurrent element in G G is periodic, and we obtain a generalization of the result of Yang [Hilbert’s fifth problem and related problems on transformation groups, American Mathematical Society, Providence, RI, 1976, pp. 142–146.] by proving that there is no nontrivial locally compact flow on R m \mathbb R^m in which all elements are recurrent.

Transcendental Julia sets with fractional packing dimension

We construct transcendental entire functions whose Julia sets have packing dimension in ( 1 , 2 ) (1,2) . These are the first examples where the computed packing dimension is not 1 1 or 2 2 . Our analysis will allow us further show that the set of packing dimensions attained is dense in the interval ( 1 , 2 ) (1,2) , and that the Hausdorff dimension of the Julia sets can be made arbitrarily close to the corresponding packing dimension.

The W. Thurston algorithm applied to real polynomial maps

This note will describe an effective procedure for constructing critically finite real polynomial maps with specified combinatorics.

Periodic points and smooth rays

Let P : C → C P: \mathbb {C} \to \mathbb {C} be a polynomial map with disconnected filled Julia set K P K_P and let z 0 z_0 be a repelling or parabolic periodic point of P P . We show that if the connected component of K P K_P containing z 0 z_0 is non-degenerate, then z 0 z_0 is the landing point of at least one smooth external ray. The statement is optimal in the sense that all but one cycle of rays landing at z 0 z_0 may be broken.

Quadrature domains and the real quadratic family

We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of the Mandelbrot set can be conformally mated with a congruence subgroup of P S L ( 2 , Z ) \mathrm {PSL}(2,\mathbb {Z}) , and that this conformal mating is the Schwarz function of a simply connected quadrature domain.

Uniformization of Cantor sets with bounded geometry

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus [Rev. Mat. Iberoamericana 15 (1999), pp. 267–277] from 2 to higher dimensions. In particular, we show that a compact subset of R n \mathbb {R}^n is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set C \mathcal {C} in R n + 1 \mathbb {R}^{n+1} .

Real entropy rigidity under quasi-conformal deformations

We set up a real entropy function h R h_\Bbb {R} on the space M d ′ \mathcal {M}’_d of Möbius conjugacy classes of real rational maps of degree d d by assigning to each class the real entropy of a representative f ∈ R ( z ) f\in \Bbb {R}(z) ; namely, the topological entropy of its restriction f ↾ R ^ f\restriction _{\hat {\Bbb {R}}} to the real circle. We prove a rigidity result stating that h R h_\Bbb {R} is locally constant on the subspace determined by real maps quasi-conformally conjugate to f f . As examples of this result, we analyze real analytic stable families of hyperbolic and flexible Lattès maps with real coefficients along with numerous families of degree d d real maps of real entropy log ⁡ ( d ) \log (d) . The latter discussion moreover entails a complete classification of maps of maximal real entropy.