Detecting linear dependence by reduction maps

We consider the local to global principle for detecting linear dependence of points in groups of the Mordell-Weil type. As applications of our general setting we obtain corresponding statements for Mordell-Weil groups of non{-}CM elliptic curves and some higher dimensional abelian varieties defined over number fields, and also for odd dimensional K-groups of number fields.Comment: This is a revised version of the MPI preprint no. 14 from March 200

Bispecial factors in circular non-pushy D0L languages

We study bispecial factors in fixed points of morphisms. In particular, we propose a simple method of how to find all bispecial words of non-pushy circular D0L-systems. This method can be formulated as an algorithm. Moreover, we prove that non-pushy circular D0L-systems are exactly those with finite critical exponent.Comment: 18 pages, 5 figure

Entropy in Dimension One

This paper completely classifies which numbers arise as the topological entropy associated to postcritically finite self-maps of the unit interval. Specifically, a positive real number h is the topological entropy of a postcritically finite self-map of the unit interval if and only if exp(h) is an algebraic integer that is at least as large as the absolute value of any of the conjugates of exp(h); that is, if exp(h) is a weak Perron number. The postcritically finite map may be chosen to be a polynomial all of whose critical points are in the interval (0,1). This paper also proves that the weak Perron numbers are precisely the numbers that arise as exp(h), where h is the topological entropy associated to ergodic train track representatives of outer automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before his death, and was uploaded by Dylan Thurston. A version including endnotes by John Milnor will appear in the proceedings of the Banff conference on Frontiers in Complex Dynamic

On reduction maps and support problem in K-theory and abelian varieties

In this paper we consider reduction maps $r_{v} : K_{2n+1}(F)/C_{F} \to K_{2n+1}(\kappa_{v})_{l}$ where $F$ is a number field and $C_{F}$ denotes the subgroup of $K_{2n+1}(F)$ generated by $l$-parts (for all primes $l$) of kernels of the Dwyer-Friedlander map and maps $r_{v} : A(F)\to A_{v}(\kappa _{v})_{l}$ where $A(F)$ is an abelian variety over a number field. We prove a generalization of the support problem of Schinzel for $K$-groups of number fields: Let $P_{1}, ..., P_{s}, Q_{1}, ..., Q_{s}\in K_{2n+1}(F)/C_{F}$ be the points of infinite order. Assume that for almost every prime $l$ the following condition holds: for every set of positive integers $m_{1}, ..., m_{s}$ and for almost every prime $v$ $$m_{1} r_{v}(P_{1})+... + m_{s} r_{v}(P_{s})=0 \mathrm{implies} m_{1} r_{v}(Q_{1})+... + m_{s}r_{v}(Q_{s})= 0.$$ Then there exist $\alpha_{i}$, $\beta_{i}\in \mathbb{Z} \setminus \{0 \}$ such that $\alpha_{i} P_{i}+\beta_{i} Q_{i}=0$ in $B(F)$ for every $i \in \{1, ... s\}$. We also get an analogues result for abelian varieties over number fields. The main technical result of the paper says that if $P_{1}, ..., P_{s}$ are nontorsion elements of $K_{2n+1}(F)/C_{F}$, which are linearly independent over $\mathbb{Z}$, then for any prime $l$, and for any set $\{k_{1},... ,k_{s}\}\subset \mathbb{N} \cup \{0\}$, there are infinitely many primes $v$, such that the image of the point $P_{t}$ via the map $r_{v}$ has order equal $l^{k_{t}}$ for every $t \in \{1, ..., s \}$.Comment: 16 page