Dynamics of Irreducible Endomorphisms of FnF_n

We consider the class non-surjective irreducible endomorphisms of the free group $F_n$. We show that such an endomorphism $\phi$ is topologically represented by a simplicial immersion $f:G \rightarrow G$ of a marked graph $G$; along the way we classify the dynamics of $\partial \phi$ acting on $\partial F_n$: there are at most $2n$ fixed points, all of which are attracting. After imposing a necessary additional hypothesis on $\phi$, we consider the action of $\phi$ on the closure $\bar{CV}_n$ of the Culler-Vogtmann Outer space. We show that $\phi$ acts on $\bar{CV}_n$ with "sink" dynamics: there is a unique fixed point $[T_{\phi}]$, which is attracting; for any compact neighborhood $N$ of $[T_{\phi}]$, there is $K=K(N)$, such that $\bar{CV}_n\phi^{K(N)} \subseteq N$. The proof uses certian projections of trees coming from invariant length measures. These ideas are extended to show how to decompose a tree $T$ in the boundary of Outer space by considering the space of invariant length measures on $T$; this gives a decomposition that generalizes the decomposition of geometric trees coming from Imanishi's theorem.Comment: v3, 46 pages, corrected gap in decomposition resul

On Invariant MASAs for Endomorphisms of the Cuntz Algebras

The problem of existence of standard (i.e. product-type) invariant MASAs for endomorphisms of the Cuntz algebra O_n is studied. In particular endomorphisms which preserve the canonical diagonal MASA D_n are investigated. Conditions on a unitary in O_n equivalent to the fact that the corresponding endomorphism preserves D_n are found, and it is shown that they may be satisfied by unitaries which do not normalize D_n. Unitaries giving rise to endomorphisms which leave all standard MASAs invariant and have identical actions on them are characterized. Finally some properties of examples of finite-index endomorphisms of O_n given by Izumi and related to sector theory are discussed and it is shown that they lead to an endomorphism of O_2 associated to a matrix unitary which does not preserve any standard MASA.Comment: 22 page

Twisted Burnside-Frobenius theory for endomorphisms of polycyclic groups

Let $R(\phi)$ be the number of $\phi$-conjugacy (or Reidemeister) classes of an endomorphism $\phi$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(\phi)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(\phi)<\infty$. Applying the result to iterations of $\phi$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ property.Comment: 11 pages, v.2: small corrections, submitte

Discrete dynamical systems in group theory

In this expository paper we describe an unifying approach for many known entropies in Mathematics. First we recall the notion of semigroup entropy h_S in the category S of normed semigroups and contractive homomorphisms, recalling also its properties. For a specific category X and a functor F from X to S, we have the entropy h_F, defined by the composition of h_S with F, which automatically satisfies the same properties proved for h_S. This general scheme permits to obtain many of the known entropies as h_F, for appropriately chosen categories X and functors F. In the last part we recall the definition and the fundamental properties of the algebraic entropy for group endomorphisms, noting how its deeper properties depend on the specific setting. Finally we discuss the notion of growth for flows of groups, comparing it with the classical notion of growth for finitely generated groups