On density of horospheres in dynamical laminations

In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both domains, starting from the famous Sullivan's no wandering domain theorem in the theory of iterations of rational functions. One of the principal objects used in the study of Kleinian groups is the hyperbolic 3- manifold associated to a Kleinian group, which is the quotient of its lifted action to the hyperbolic 3- space. M.Lyubich and Y.Minsky have suggested to extend Sullivan's dictionary by providing an analogous construction for iterations of rational functions. Namely, they have constructed a lamination by three-dimensional manifolds equipped with a continuous family with hyperbolic metrics on them (may be with singularities). The action of the rational mapping on the sphere lifts naturally up to homeomorphic action on the hyperbolic lamination that is isometric along the leaves. The action on the hyperbolic lamination admits a well-defined quotient called {\it the quotient hyperbolic lamination}. We study the arrangement of the horospheres in the quotient hyperbolic lamination. The main result says that if a rational function does not belong to a small list of exceptions (powers, Chebyshev and Latt\`es), then there are many dense horospheres, i.e., the horospheric lamination is topologically-transitive. We show that for "many" rational functions (hyperbolic or critically-nonrecurrent nonparabolic) the quotient horospheric lamination is minimal: each horosphere is dense.Comment: The complete versio

Global Solutions vs. Local Solutions for the AI Safety Problem

There are two types of artificial general intelligence (AGI) safety solutions: global and local. Most previously suggested solutions are local: they explain how to align or “box” a specific AI (Artificial Intelligence), but do not explain how to prevent the creation of dangerous AI in other places. Global solutions are those that ensure any AI on Earth is not dangerous. The number of suggested global solutions is much smaller than the number of proposed local solutions. Global solutions can be divided into four groups: 1. No AI: AGI technology is banned or its use is otherwise prevented; 2. One AI: the first superintelligent AI is used to prevent the creation of any others; 3. Net of AIs as AI police: a balance is created between many AIs, so they evolve as a net and can prevent any rogue AI from taking over the world; 4. Humans inside AI: humans are augmented or part of AI. We explore many ideas, both old and new, regarding global solutions for AI safety. They include changing the number of AI teams, different forms of “AI Nanny” (non-self-improving global control AI system able to prevent creation of dangerous AIs), selling AI safety solutions, and sending messages to future AI. Not every local solution scales to a global solution or does it ethically and safely. The choice of the best local solution should include understanding of the ways in which it will be scaled up. Human-AI teams or a superintelligent AI Service as suggested by Drexler may be examples of such ethically scalable local solutions, but the final choice depends on some unknown variables such as the speed of AI progres