TITLE: What are some examples of "chimeras" in mathematics?
QUESTION [25 upvotes]: The best example I can think of at the moment is Conway's surreal number system, which
combines 2-adic behavior in-the-small with $\infty$-adic behavior in the large.  The surreally 
simplest element of a subset of the positive (or negative) integers is the one closest to 0 
with respect to the Archimedean norm, while the surreally simplest dyadic rational in a 
subinterval of (0,1) (or more generally $(n,n+1)$ for any integer $n$) is the one closest to 0 
with respect to the 2-adic norm (that is, the one with the smallest denominator).  
This chimericity also comes up very concretely in the theory of Hackenbush strings: 
the value of a string is gotten by reading the first part of the string as the unary representation 
of an integer and the rest of the string as the binary representation of a number between 
0 and 1 and adding the two.  
I'm having a hard time saying exactly what I mean by chimericity in general, but 
some non-examples may convey a better sense of what I don't mean by the term.
A number system consisting of the positive reals and the negative integers would
be chimeric, but since it doesn't arise naturally (as far as I know), it doesn't qualify.
Likewise the continuous map from $\bf{C}$ to $\bf{C}$ that sends $x+iy$ to $x+i|y|$ is chimeric (one does 
not expect to see a holomorphic function and a conjugate-holomorphic function stitched 
together in this Frankenstein-like fashion), so this would qualify if it ever arose naturally, but I've never seen anything like it.
Non-Euclidean geometries have different behavior in the large and in the small, but the
two behaviors don't seem really incompatible to me (especially since it's possible to
continuously transition between non-zero curvature and zero curvature geometries).
One source of examples of chimeras could be physics, since any successful Theory Of 
Everything would have to look like general relativity in the large and quantum theory in
the small, and this divide is notoriously difficult to bridge.  But perhaps there are other 
mathematical chimeras with a purely mathematical genesis.
See also my companion post Where do surreal numbers come from and what do they mean? .

REPLY [4 votes]: 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 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.