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Register a new userAny counterexample to Makienkos conjecture is an indecomposable continuum
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Makienkos conjecture, a proposed addition to Sullivans dictionary, can be stated as follows: The Julia set of a rational function R has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienkos conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational functions whose Julia set is an indecomposable continuum.
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We prove the entropy conjecture of M. Barge from 1989: for every $rin [0,infty]$ there exists a pseudo-arc homeomorphism $h$, whose topological entropy is $r$. Until now all pseudo-arc homeomorphisms with known entropy have had entropy $0$ or $infty$.
We prove that the Laptev--Safronov conjecture (Comm. Math. Phys., 2009) is false in the range that is not covered by Franks positive result (Bull. Lond. Math. Soc., 2011). The simple counterexample is adaptable to a large class of Schrodinger type operators, for which we also prove new sharp upper bounds.
Answering a question of P. Bankston, we show that the pseudoarc is a co-existentially closed continuum. We also show that $C(X)$, for $X$ a nondegenerate continuum, can never have quantifier elimination, answering a question of the the first and third named authors and Farah and Kirchberg.
Payne conjectured in 1967 that the nodal line of the second Dirichlet eigenfunction must touch the boundary of the domain. In their 1997 breakthrough paper, Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili proved this to be false by constructing a counterexample in the plane with many holes and raised the question of the minimum number of holes a counterexample can have. In this paper we prove it is at most 6.
The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.
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