IA Scholar Query: Applications of pcf for mild large cardinals to elementary embeddings.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 16 Dec 2021 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Ahlfors-regular conformal dimension and energies of graph maps
https://scholar.archive.org/work/dzicvzm3vzcidnng65s5n2rzma
For a hyperbolic rational map f with connected Julia set, we give upper and lower bounds on the Ahlfors-regular conformal dimension of its Julia set J_f from a family of energies of associated graph maps. Concretely, the dynamics of f is faithfully encoded by a pair of maps π, ϕ : G_1 → G_0 between finite graphs that satisfies a natural expanding condition. Associated to this combinatorial data, for each q ≥ 1, is a numerical invariant E^q[π,ϕ], its asymptotic q-conformal energy. We show that the Ahlfors-regular conformal dimension of J_f is contained in the interval where E^q=1. Among other applications, we give two families of quartic rational maps with Ahlfors-regular conformal dimension approaching 1 and 2, respectively.Kevin M. Pilgrim, Dylan P. Thurstonwork_dzicvzm3vzcidnng65s5n2rzmaThu, 16 Dec 2021 00:00:00 GMTGalois groups and prime divisors in random quadratic sequences
https://scholar.archive.org/work/5gnvhbo4eraujobssyffvm6moe
Given a set S={x^2+c_1,...,x^2+c_s} with c_i∈ℤ and an infinite sequence γ of elements of S, one can associate an arboreal representation to γ, analogous to the case of iterating a single polynomial. Under suitable conditions, we conjecture that a positive proportion of these sequences furnish large image representations. As evidence, we classify all S with a particular obstruction and prove a version of this conjecture over ℤ[t]. Finally as an application of large image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences.Vivian Olsiewski Healey, Wade Hindes, Rafe Joneswork_5gnvhbo4eraujobssyffvm6moeWed, 25 Aug 2021 00:00:00 GMT