Fully irreducible automorphism

In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group "Fn" is an element of Out("Fn") which has no periodic conjugacy classes of proper free factors in "Fn" (where "n" > 1). Fully irreducible automorphisms are also referred to as "irreducible with irreducible powers" or "iwip" automorphisms. The notion of being fully irreducible provides a key Out("Fn") counterpart of the notion of a pseudo-Anosov element of the mapping class group of a finite type surface. Fully irreducibles play an important role in the study of structural properties of individual elements and of subgroups of Out("Fn").
Formal definition.
Let formula_1 where formula_2. Then formula_3 is called "fully irreducible" if there do not exist an integer formula_4 and a proper free factor formula_5 of formula_6 such that formula_7, where formula_8 is the conjugacy class of formula_5 in formula_6. 
Here saying that formula_5 is a proper free factor of formula_6 means that formula_13 and there exists a subgroup formula_14 such that formula_15.
Also, formula_16 is called "fully irreducible" if the outer automorphism class formula_1 of formula_18 is fully irreducible.
Two fully irreducibles formula_19 are called "independent" if formula_20.
Relationship to irreducible automorphisms.
The notion of being fully irreducible grew out of an older notion of an ``irreducible" outer automorphism of formula_6 originally introduced in. An element formula_1, where formula_2, is called "irreducible" if there does not exist a free product decomposition
with formula_25, and with formula_26 being proper free factors of formula_6, such that formula_3 permutes the conjugacy classes formula_29.
Then formula_1 is fully irreducible in the sense of the definition above if and only if for every formula_4 formula_32 is irreducible.
It is known that for any "atoroidal" formula_1 (that is, without periodic conjugacy classes of nontrivial elements of formula_6), being irreducible is equivalent to being fully irreducible. For non-atoroidal automorphisms, Bestvina and Handel produce an example of an irreducible but not fully irreducible element of formula_35, induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.
Properties.
Moreover, formula_49 is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of formula_3.