Digest of Doctral thesis
TITLE: The rotation sets versus the Markov partitions
ABSTRACT :
The theory of the rotation set is diligently studied.
Newhouse, Palis, Takens firstly defined the rotation set
(this case, rotation interval) of
circle maps of degree one, which is a direct extension of the rotation number
of a circle homeomorphism by Poincare.
The rotation set of continuous maps of higher dimensional torus homotopic to the
identity is defined by
Kim, Mackey and Guckenheimer, Llible and Mackey etc.,
and, Misiurewicz and Ziemian
digested them systematically.
The rotation set of 2 dimensional surface with higher genus homeomorphisms
homotopic to the identity is defined by Pollicott
and by Franks. These two definitions are slightly different.
In this paper we define the rotation set of homeomorphisms f of
a closed manifold M
whose associated homomorphism f* on H1(M; Z)
is the identity by an extension of the definition of Franks.
On the other hand, the symbolic dynamics (or shift automorphisms)
is also studied diligently.
Particularly the theory of the dynamics of the subshift of finite type
is closely related with the theory of Markov partitions.
If (M,f) has Markov partitions,
the dynamics of (M,f) is translated into that of a subshift of finite type.
The first example of the Markov partition was found by Adler and Weiss for
an Anosov diffeomorphism on 2-dimensional torus.
The systematic introduction of Markov partitions is due
to Sinai for Anosov diffeomorphisms.
Bowen construct Markov partitions on basic sets of Axiom A diffeomorphisms.
In this paper we define the rotation set of
homeomorphisms f whose associated homomorphism
f* on H1(M; Z) is the identity, and
we show the relation between the rotation set
and the Markov partition.
We give the way to calculate the rotation set from the Markov partition
and show that the rotation set is a convex polygon and explicit representation of
every extremal point of this polygon.
Our main results are :
Theorem
Let (M,f) be a homeomorphism whose associated homomorphism f*
on H1(M; Z) is the identity map, and suppose (M,f) has a Markov partition
R= { Ri }i of M.
Let { ƒ¿i,j }i,j
be the set of transition loops for (M,f, R, O) and
let [ƒ¿i,j] be the homology class of ƒ¿i,j.
Suppose the itinerary I(x) of x in M is
I(x) = ... ,i-2,i-1,i0,i1,i2,... and
the image of the 2-block map S(I(x)) of I(x) is
S(I(x))=... [ƒ¿i-2,i-1][ƒ¿i-1,i0]
[ƒ¿i0,i1]
[ƒ¿i1,i2]...
Then the rotation vector ƒÏ(x,f) of x is given by
ƒÏ(x,f)= ƒ°i,j P(ƒ¿i,j) [ƒ¿i,j]
where P(ƒ¿i,j) is
the appearance probability of the subsequence "i, j"
in the itinerary
I(x)=... ,i-2,i-1,i0,i1,i2,...
of x if P(ƒ¿i,j) exists.
and
Theorem
Let (M,f) be a transitive homeomorphism whose associated homomorphism f*
on H1(M;\Bbb Z) is the identity map and suppose (M,f) has a Markov partition
R= {Ri} of M.
Then the rotation set Rot(f)
is a finite polygon and every extremal point is obtained by
the rotation vector of some periodic point.
Bowen constructed Markov partitions of the basic sets of the Axiom A
diffeomorphisms, so we conclude that the rotation set of Axiom A diffeomorphisms
with f*=id
is a finite polygon if f is restricted on one basic set.
Since the omega limit set for every x on M
is contained in the nonwandaring set,
we have only to focus on the nonwandering sets and we have
Corollary
For an Axiom A diffeomorphism f with f*=id,
the rotation set is a finite union of finite polygons.
Thurston defined the pseudo Anosov diffeomorphisms and constructed Markov partitions.
Thus we have
Corollary
For a pseudo Anosov diffeomorphism f with f*=id,
the rotation set is a finite polygon.
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