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For any $ delta >0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=delta$. Stallard proved that the dimension must be strictly larger than 1 whenever $f$ has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.
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We show that for each $din (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.
If $X$ is the attractor set of a conformal IFS in dimension two or three, we prove that there exists a quasiregular semigroup $G$ with Julia set equal to $X$. We also show that in dimension two, with a further assumption similar to the open set condition, the same result can be achieved with a semigroup generated by one element. Consequently, in this case the attractor set is quasiconformally equivalent to the Julia set of a rational map.
Let $f : Xto X$ be a dominating meromorphic map on a compact Kahler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0leq l leq k$. For an ergodic probability measure $ u$, we extend similarly the notion of measure-theoretic entropy $h_{ u}^l(f)$. Under mild hypothesis, we compute $h^l_{mathrm{top}}(f)$ in term of the dynamical degrees of $f$. In the particular case of endomorphisms of $mathbb{P}^2$ of degree $d$, we show that $h^1_{mathrm{top}}(f)= log d$ for a large class of maps but we give examples where $h^1_{mathrm{top}}(f) eq log d$.
We construct the first examples of rational functions defined over a non-archimedean field with certain dynamical properties. In particular, we find such functions whose Julia sets, in the Berkovich projective line, are connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we show for some of our examples that the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.
We give an introduction to buried points in Julia sets and a list of questions about buried points, written to encourage aficionados of topology and dynamics to work on these questions.
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