Discrete Holomorphic Local Dynamical Systems

76 Eric Bedford
Fig. 4 Slice W u (q β ) ∩ K +
Problem Suppose that J is connected, and f is hyperbolic. How do you write J as a
quotient of the solenoid? What quotients can appear?
Notes. The geometry of U + is discussed further in [HO]. The problem of writing
J as a quotient of the solenoid is discussed in [BS7]. The thesis of Oliva [O] gives
a number of computer experiments that are helpful in understanding what identifications
can appear when writing J as a quotient. The papers [I] and [IS] give a
combinatorial/topological approach to this problem.
1.7 Currents on R N
Let D k be the set of smooth k-forms on RN with compact support. We define the
space of k-currents to be the dual: D ′ k :=(Dk ) ′ . A basic example is the current of integration.LetM⊂
RN be a k-dimensional submanifold which is oriented and has locally
bounded (k-dimensional) area. Then the current of integration [M] is defined by
D k �
∋ ϕ ↦→〈[M],ϕ〉 = ϕ.
If ϕ is a k-form, and if t is a k-vector, then x ↦→ ϕ(x)·t is a scalar-valued function.
Now let ν be a Borel measure on RN ,andlett(x) denote a field of k-vectors on RN which is Borel measurable, and which is locally integrable with respect to ν. Then
the current T = tν is defined by the action
D k �
∋ ϕ ↦→〈T,ϕ〉 := ϕ(x) · t(x)ν(x).
M