Related papers: Computability properties of hyperbolic complex H\'…

We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C^2. We develop several aspects of the question, which was first studied by Bedford-Smillie. We introduce a new sufficient condition for the…

Hyperbolic spaces, which have the capacity to embed tree structures without distortion owing to their exponential volume growth, have recently been applied to machine learning to better capture the hierarchical nature of data. In this…

Hyperbolic deep learning leverages the metric properties of hyperbolic spaces to develop efficient and informative embeddings of hierarchical data. Here, we focus on the solvable group structure of hyperbolic spaces, which follows naturally…

We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is…

There are two natural definitions of the Julia set for complex H\'enon maps: the sets $J$ and $J^\star$. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts…

We consider the structure of substantially dissipative complex H\'enon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points…

Let $g$ be a polynomial automorphism of $\C^2$. We study the Hausdorff dimension and topological dimension of the Julia set of $g$. We show that when $g$ is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly…

In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case…

We give a topological model of the critical locus for complex H\'{e}non maps that are perturbations of the quadratic polynomial with disconnected Julia set.

We completely characterize the conformal radii of Siegel disks in the family $$P_\theta(z)=e^{2\pi i\theta}z+z^2,$$ corresponding to {\bf computable} parameters $\theta$. As a consequence, we constructively produce quadratic polynomials…

Many high-dimensional practical data sets have hierarchical structures induced by graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform…

For any integers $d\ge 3$ and $n\ge 1$, we construct a hyperbolic rational map of degree $d$ such that it has $n$ cycles of the connected components of its Julia set except single points and Jordan curves.

Hyperbolic geometry has emerged as a powerful tool for modeling complex, structured data, particularly where hierarchical or tree-like relationships are present. By enabling embeddings with lower distortion, hyperbolic neural networks offer…

We discuss computability of impressions of prime ends of compact sets. In particular, we construct quadratic Julia sets which possess explicitly described non-computable impressions.

The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres and closed manifolds: if a set $X$ is homeomorphic to a sphere or a closed manifold, then any…

This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian…

We establish upper bounds of bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs. Here we continue the research started in in our revious publications where computability, in the rigorous sense of…

Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of complex H\'enon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have a semi-parabolic fixed point with one eigenvalue $\lambda=e^{2\pi i p/q}$. We give…

In this note we give answers to questions posed to us by J.Milnor and M.Shub, which shed further light on the structure of non-computable Julia sets.

We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.