We consider cubic polynomials $f(z)=z^3+az+b$ defined over $\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier $\lambda$. In the case $N=1$, there are infinitely many such objects, and in the case $N\geq 3$, only finitely many (subject to a mild assumption). The case $N=2$ has particularly rich structure, and we are able to describe all such cubic polynomials defined over the field $\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.

We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another: heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface, and do not interact with each other. We consider a generalization of Newton's minimal resistance problem: given two bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2 \subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the resistance in the class of connected bodies $B$ such that $C_1 \subset B \subset C_2$. We prove that the infimum of resistance is zero; that is, there exist "almost perfectly streamlined" bodies.

In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$.

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Real s| \leq K$, where $\delta$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott--Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\{|\Real s | \leq |\Imag s|^\alpha\}$ is given, followed by weaker lower bound estimates in strips $\{\Real s > -C, |\Imag s|\leq r\}$, and logarithmic neighbourhoods $\{ |\Real s | \leq \rho \log |\Imag s| \}$. Recent numerical work of Strain--Zworski suggests the upper bounds in strips are optimal.