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Outer automorphisms of free groups have a chimeric nature, somewhat like mapping classes of surfaces but with weirder pieces and stranger stiches. The pieces are strata of relative train track maps'', and are somewhat analogous to the subsurfaces of the Thurston decomposition of a mapping class, but the strata can spill over and interact with each other in ways that the subsurfaces cannot.

For instance, you can have two different exponentially growing strata, which as in the surface situation correspond to two different exponenially stretched laminations each having a dense leaf, but one of those laminations contains the other as a sublamination.

You can also have an exponentially growing stratum and a fixed stratum --- the latter corresponding analogous to a subsurface on which the mapping class is the identity --- but the lamination corresponding to the exponentially growing stratum scribbles all over the fixed stratum, filling it up with junk.

And then there are the linearly and polynomially growing strata. A linear stratum spills over a fixed stratum, a quadratic stratum spills over a linear stratum, etc. And last but not least, there are the nonexponentially growing strata that spill over exponentially growing strata; I still can't decide whether they grow or they don't grow under iteration of the outer automorphism.

2 added 42 characters in body

Outer automorphisms of free groups have a chimeric nature, somewhat like mapping classes of surfaces but with weirder pieces and stranger stiches. The pieces are strata of relative train track maps'', and are somewhat analogous to the subsurfaces of the Thurston decomposition of a mapping class, but the strata can spill over and interact with each other in ways that the subsurfaces cannot.

For instance, you can have two different exponentially growing strata, which as in the surface situation correspond to two different exponenially stretched laminations each having a dense leaf, but one of those laminations contains the other as a sublamination.

You can also have an exponentially growing stratum and a fixed stratum --- the latter corresponding to a subsurface on which the mapping class is the identity --- but the lamination corresponding to the exponentially growing stratum scribbles all over the fixed stratum, filling it up with junk.

And then there are the linearly and polynomially growing strata. A linear stratum spills over a fixed stratum, a quadratic stratum spills over a linear stratum, etc. And last but not least, there are the nonexponentially growing strata that spill over exponentially growing strata; I still can't decide whether they grow or they don't grow under iteration of the outer automorphism.

Outer automorphisms of free groups have a chimeric nature, somewhat like mapping classes of surfaces but with weirder pieces and stranger stiches. The pieces are strata of relative train track maps'', and are somewhat analogous to the subsurfaces of the Thurston decomposition of a mapping class, but the strata can spill over and interact with each other in ways that the subsurfaces cannot.