Alon Levy

My research interests are in arithmetic dynamical systems, invariant theory and moduli spaces, and p-adic analysis and geometry. My email is alonlevy@kth.se.

Below are links to preprints of my papers. If they conflict with the arXiv citations, the versions here are more current.

1. The Space of Morphisms on Projective Space, Acta Arithmetica 146 (2011), 13-31
2. The Semistable Reduction Problem for the Space of Morphisms on Pⁿ, Algebra and Number Theory 6 No. 7 (2012), 1483-1501
3. Attracting Cycles in p-adic Dynamics and Height Bounds for Post-Critically Finite Maps, joint with Rob Benedetto, Patrick Ingram, and Rafe Jones, Duke Math Journal 163 No. 13 (2014), 2325-2356
4. An Algebraic Proof of Thurston's Rigidity for Maps With a Superattracting Cycle, submitted
5. Uniform Bounds for Pre-Periodic Points in Families of Twists, joint with Michelle Manes and Bianca Thompson, Proceedings of the American Mathematical Society 142 (2014), 3075-3088
6. Isotriviality and the Space of Morphisms on Projective Varieties, joint with Anupam Bhatnagar, submitted
7. The McMullen Map in Positive Characteristic, submitted
8. Finite Ramification for Preimage Fields of Postcritically Finite Morphisms, joint with Andrew Bridy, Patrick Ingram, Rafe Jones, Jamie Juul, Michelle Manes, Simon Rubinstein-Salzedo, and Joseph Silverman, submitted
9. Isolated Periodic Points in Several Nonarchimedean Variables, submitted
10. Eventually Stable Rational Functions, joint with Rafe Jones, to appear in International Journal of Number Theory (2016).

One more paper is in pre-preprint form; text is available to colleagues upon request:

• PCF All the Way Down, joint with Tom Tucker. We prove that if f: X --> X is postcritically finite, then the restriction of the appropriate iterate of f to any periodic hypersurface is also postcritically finite. We use this to strengthen the main theorem in Bridy et al.

Medium-term projects, with submission expected by early 2017:

• Eventually Stable Quadratic Functions, joint with John Doyle and Tom Tucker. We use the eventual stability result above of myself and Jones to prove that quadratics defined over a function field of characteristic zero are eventually stable.
• Berkovich Space in Several Variables, joint with Tyler Foster. We come up with an explicit description of a large slice of Berkovich Pⁿ, suitable for doing dynamics, by analyzing all the possible images of polydisks under morphisms from Pⁿ to itself. Our tools include tropical geometry and higher-dimensional rigid geometry.

Longer-term projects:

• The abc Conjecture and Arboreal Representations, joint with Rafe Jones. We define the arboreal representation to be the action of the absolute Galois group G_K on the infinite backward image tree of a in K under f: P¹_K --> P¹_K. Conjecturally, it acts on the tree with finite-index image when the critical orbits of f are infinite and independent. We study this question under the assumptions that the abc conjecture holds and that f is eventually stable.
• Attracting Cycles in Higher-Dimensional Non-Archimedean Dynamics. I use higher-dimensional rigid geometry to generalize the attracting cycles result with Benedetto, Ingram, and Jones to self-morphisms of Pⁿ. Key tools include my in-preparation papers above with Tucker and with Foster.

2016-11-12 18:05 CET