Iterated monodromy groups of the tent map
Volodymyr Nekrashevych
March 14, 2013
Les Diablerets
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Iterated monodromy group
Let f : M1 −→ M be a finite covering map, where M
Then the
F1 ⊂ M.
−n
fundamental group π1 (M, t) acts on the rooted tree n≥0 f (t) by the
monodromy actions on the levels.
The quotient of π1 (M, t) by the kernel of the action is the iterated
monodromy group IMG(f ).
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Post-critically finite rational function
Let f (z) ∈ C(z) be such that the forward orbits of the critical points of f
are finite. Let Pf be the union of the orbits of the critical values. Then
b \ f −1 (Pf ) ⊂ M := C
b \ Pf , and f : M1 −→ M is a covering.
M1 := C
Example: z 2 + i . Its critical points are 0 and ∞. The orbits of the critical
points are 0 7→ i 7→ −1 + i 7→ −i 7→ −1 + i , and ∞ 7→ ∞.
The iterated monodromy group IMG(z 2 + i ) is generated by
a(0w ) = 1w ,
a(1w ) = 0w
b(0w ) = 0c(w ),
b(1w ) = 1w
c(0w ) = 0a(w ),
c(1w ) = 1b(w )
We write
a = σ,
V. Nekrashevych (Texas A&M)
b = (c, 1),
c = (a, b).
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Self-similar groups
A group G acting faithfully on X∗ is self-similar if for all g ∈ G and x ∈ X
there exist h ∈ G and y ∈ X such that
g (xw ) = yh(w )
for all w ∈ X∗ .
Iterated monodromy groups have natural self-similar actions.
If h is shorter than g for long g , then G is said to be contracting.
Iterated monodromy groups of expanding (sub-hyperbolic) maps are
contracting.
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Schreier graphs
Let G be a group acting by automorphisms on the tree X∗ . Let S = S −1
be a finite generating set. (Simplicial) Schreier graphs Γn (G , S) are the
graphs with the set of vertices X n in which two vertices v1 , v2 belong to
one edge if and only if g (v1 ) = v2 for some g ∈ S.
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The Schreier graphs of IMG(f ) approximate the Julia set of f .
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The Schreier graphs of IMG(f ) approximate the Julia set of f .
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Limit space
Let G be a contracting self-similar group acting on the tree of words over
an alphabet X. Consider the space X−ω of left-infinite sequences of
elements of X with the direct power topology. We say that . . . x2 x1 and
. . . y2 y1 are equivalent with respect to G if xn . . . x2 x1 and yn . . . y2 y1 are
on a uniformly bounded distance from each other in the Schreier graphs of
the action of G on Xn for all n.
The quotient of X−ω is the limit space JG . The shift . . . x2 x1 7→ . . . x3 x2
induces a continuous map s : JG −→ JG .
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Theorem
If f : M −→ M is an expanding self-covering of a compact connected
metric space, then (M, f ) is topologically conjugate to the limit dynamical
system of IMG(f ).
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Orbispaces
Some post-critically finite rational functions are sub-hyperbolic, i.e., the
post-critical set Pf intersects with the Julia set (e.g. z 2 + i ).
We consider in this case a pair of maps f : M1 −→ M and
ι : M1 −→ M, where M and M1 are orbispaces, f is a covering, and ι is
an embedding.
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Orbispaces
An orbispace M is an equivalence class of a proper pseudogroup (with
Hausdorff groupoid of germs). The pseudogroup is pseudogroup of
changes of charts in the corresponding atlas.
Neighborhood of every point of M is represented as a quotient of a
topological space by an action of a finite group (the isotropy group of the
point).
An embedding of orbispaces is defined by an open full functor of the
groupoids of germs of the pseudogroups of changes of charts. In
particular, it is surjective on isotropy groups.
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General question
Given a topological dynamical system (J , f ) find all self-similar group
whose limit dynamical system (JG , s) is topologically conjugate to (J , f ).
If f : J −→ J is an expanding self-covering of a compact connected
metric space, then the only such group is IMG(f ).
If f : J −→ J is sub-hyperbolic, then such groups are the iterated
monodromy groups of pairs p : M1 −→ M, ι : M1 −→ M, where M1
and M are orbispace structures on J ; p = f and ι = id, if we erase the
orbispace structure.
If f (z) is a p.c.f. rational function, then there are natural orbispace
structures M, M1 on the Julia set of f , defined by W. Thurston. The
isotropy groups are cyclic. It corresponds to computation of IMG(f ) using
the punctured sphere, as above.
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The tent map
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What are the iterated monodromy groups of the tent map? In other
words, what self-similar (self-replicating) groups have the limit dynamical
system which is topologically conjugate to the tent map.
Non-trivial isotropy groups are only at the post-critical points, which are
the endpoints in this case. Thus, M and M1 are graphs of two groups
connected by an edge (with trivial edge-group).
It follows that H ∼
= C2 . Let φ : G −→ G and ψ : G −→ H be the
epimorphisms induced by ι. Then IMG(f ) is generated by G and
H = {1, σ} subject to recursion
g = (ψ(g ), φ(g ))
for all g ∈ G .
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The image of G in IMG(f ) is isomorphic to C2n for some n. Let us identify
G with the vector space Fn2 . We may assume that ψ is given by the matrix
(0, 0, . . . , 0, 1), and φ is given by a matrix


0 0 . . . 0 a0
 1 0 . . . 0 a1 


 0 1 . . . 0 a2 

.
 .. .. . . ..
.. 
 . .
. .
. 
0 0 . . . 1 an−1
Here p(x) = x n + an−1 x n−1 + · · · + a0 is the characteristic polynomial of
φ. We will denote the corresponding iterated monodromy group by Gp(x) .
The class of groups Gp(x) was defined by Z. Šunić in 2002.
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The “smallest” case of an iterated monodromy group of the tent map
comes from the most natural choice G = C2 and the identical maps φ and
ψ, i.e., for p(x) = x + 1. This is the only case when ι is an isomorphism.
It is also the iterated monodromy group of the polynomial z 2 − 2.
It is generated by:
a = σ,
b = (a, b).
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The Schreier graphs Γn (Gp(x) , S) are chains of 2n vertices.
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The next examples are Gx 2 +1 and Gx 2+x+1 . Both of them are Grigorchuk
groups. Gx 2 +x+1 is the Grigorchuk group. Gx 2 +1 is the group G0101... from
the uncountable family of Grigorchuk groups.
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x2
0 1
1 1
. Denote
+ x + 1. Then φ is given by the matrix
0
1
1
a = σ ∈ H, b =
, c = φ(b) =
, and d = φ(c) =
.
1
1
0
Then ψ(b) = ψ(c) = a, and ψ(d) = 1. It follows that a, b, c, d are defined
by:
a = σ, b = (a, c), c = (a, d), d = (1, b).
Let p(x) =
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Properties of the groups Gp(x)
Theorem (Z. Šunić)
If p1 (x)|p2 (x), then Gp1 (x) ≤ Gp2 (x) . If p(x) is not divisible by x + 1, then
Gp(x) is a 2-group. If p(x) 6= x + 1, then Gp(x) has intermediate growth.
Theorem (Y. Bondarenko, D. D’Angeli, T. Nagnibeda)
If G is a group generated by bounded automata over the binary alphabet
such that almost all Schreier graphs of the action of G on the boundary of
the tree have two ends, then G is isomorphic to Gp(x) for some p(x).
Are Gp(x) , their analogs for heigher degrees, and the adding machine
actions the only contracting self-similar groups whose Schreier graphs have
two ends?
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Kerchief folding
A 2-dimensional analog of the tent map is the map folding an isoceles
right triangle in two:
Another analog is the “A4 paper”.
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Similarly to the tent map, there is a natural orbispace structure on the
triangle, making the map a self-covering of an orbispace. An atlas is R2
with the action of the group of all isometries of Z2 .
The iterated monodromy group is the group of all isometries of Z2 , and is
an extension of Z2 by D4 . It is generated by
a = σ,
b = (c, c),
c = (a, b)
The isotropy groups: GA ∼
= GB ∼
= D4 , GC ∼
= D2 , the edge groups are C2 .
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Consider groups generated by three groups GAC ∼
= C2n1 , GAB = C2n2 , and
GBC = {1, a} ∼
= C2 defined by the recursions:
a = σ,
g1 = (φ0 (g1 ), φ1 (g1 )),
g2 = (ψ0 (g2 ), ψ1 (g2 )),
for g1 ∈ GAC , g2 ∈ GAB , where φ0 , φ1 : GAC −→ GAB , ψ0 : GAB −→ GBC ,
ψ1 : GAB −→ GAC are homomorphisms.
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Schreier graphs of all such groups are sub-graphs of the Schreier graphs of
the virtually abelian iterated monodromy group of the kerchief folding
map.
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IMG(z 2 + i ) = ha = σ, b = (c, 1), c = (a, b)i
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G0101... = ha = σ, b = (1, c), c = (a, b)i
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′
′ ≤G
If there exist subgroups GAC
AC and GAB ≤ GAB such that restrictions
of φi and ψi are isomorphisms, then the group is the iterated monodromy
group of a uniformization of the kerchief folding.
Complete description of the iterated monodromy groups of kerchief folding
is obtained by increasing the vertex groups GA , GB , GC .
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Grigorchuk group
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A “two-dimensional Grigorchuk group”
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Applications to dynamics
If G1 and G2 are contracting self-similar groups acting on X∗ such that
G1 < G2 , then the embedding induces a semi-conjugacy λ : JG1 −→ JG2
of the limit dynamical systems, since the Schreier graphs of G1 are
subgraphs of the Schreier graphs of G2 .
s
JG1 −→

yλ
s
JG1

yλ
JG2 −→ JG2 .
All semi-conjugacies between sub-hyperbolic dynamical systems are
induced by morphisms of self-similar groups.
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Carateodori loop
Let f be a monic p.c.f. polynomial of degree d. The action of f near
infinity is the same as the action of z d . This defines an embedding
IMG(z d ) −→ IMG(f ), which induces a surjective semi-conjugacy
R/Z −→ Jf called the Carateodory loop.
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Mating
Let f and g be p.c.f. polynomials of degree d. Paste together the filled-in
Julia sets of f and g along the Carateodory loop (but flipping the loop
using complex conjugation). The resulting dynamical system is the mating
of f and g .
An iterated monodromy group of the mating is generated by IMG(f ) and
IMG(g ). The coverse problem of deciding if a given dynamical system
(e.g. a rational function) is a mating involves finding a uniformization of
the system such that the corresponding i.m.g. is generated by two i.m.g.’s
of polynomials.
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