From K to L

• \( \mathcal{K} = \{\phi_i :A_i \to B_i \}_{i=1, … , k} \) be a non-nesting closed system on a finite tree with at least one infinite orbit.
• Non nesting forces a special structure for the set of finite regular \( \mathcal{K} \) – orbits
  1. Union of all finite regular orbits = a finite union of open intervals
  2. Its complement \( K_1 \) is a finite union of closed subtrees, not all of them points.
• Disregard isolated points of \( K_1 \)
• D = disjoint union of all closed edges of \( K_1 \)
• D is a multi – interval (a finite disjoint union of compact intervals)
• \( \delta D \) = set of endpoints of compact intervals in D.
• int D = \( D \backslash \delta D\)
• The system of maps \( \mathcal{K} \) naturally induces a system \( \mathcal{L} \) on D
  • Replace each \( \phi_i \) by the collection of its restrictions to edges of \( K_i \), keeping only those maps whose domains contains more than a single point.
  • We may split domains at preimages of vertices of \( B_i \)
• The system \( \mathcal{L} \) is closed and non-nesting
• Every regular orbit of \( \mathcal{L} \) is infinite
• Every \( \mathcal{L} \) invariant probability measure without atoms corresponds to a \( \mathcal{K} \) invariant probability measure (supported on \( K_1\) )
• Hence it is sufficient to find an \( \mathcal{L} \) invariant measure

From L to M

• Let \( \overset{o}{\mathcal{L}} \) be the open system obtained by replacing each element \( \phi : A \to B \) of \( \mathcal{L} \) to the open interval A \ {endpoints}.
• Hardest case: every minimal closed(?) \( \overset{o}{\mathcal{L}} \) invariant set is a finite singular orbit (we need dense orbits).
  • Why is this hardest? What is singular orbit?
  • We want every orbit to be dense (and the system to be open) to apply Sacksteders theorem.
  • In the worst case, none-of the orbits are dense. Since they are infinite, there must be accumulation points (every infinite sequence has a convergent subsequence). These accumulation points could be inside or at the end point
  • If we look at the accumulation points they form a minimal invariant set. If they are finite and singular, it is hardest to get and infinite dense orbit out of them.
• There are finitely many minimal sets, among which are the endpoints of \( d \in \delta D \)
• By splitting D (thus changing \( \mathcal{L} \) and \( \overset{o}{\mathcal{L}} \), we may assume that the points of \( \delta D \) are the only minimal sets of \( \overset{o}{\mathcal{L}} \)
• Note that the closure of every \( \overset{o}{\mathcal{L}} \)-orbit contains an end point of D.
• For every \( d \in \delta D \), we choose an \( \mathcal{L} \) word \( w_d \) defined on a non – degenerate closed interval \( I_d \) containing d and sending d to a point \( \tilde {d} \in int D \)
  • Such a word exists because otherwise non-nesting of \( \mathcal{L} \) would imply finiteness of orbits near d.
• \( E = \{ p \in \delta D\} \) such that some \( \overset{o}{\mathcal {L}}\) – orbit accumulates on p.
  • This implies we can get points of some orbit arbitrarily close to p.
  • Choose \( x_p , y_p \) from these points (see next).
• For each \( p \in E \) choose \( x_p, y_p \in I_p \) such that some \( \overset{o}{\mathcal {L}}\) – word \( \tau_p \) sends \( x_p \) to \( y_p \) in an orientation preserving way.
• Since every \( \overset{o}{\mathcal {L}}\) – orbit accumulates on E, we may require, furthermore, that for every \( d \in \delta D \) the orbit of \( \tilde{d} \) meets the union of all intervals \( (x_p , y_p ), p \in E \).
  • Notice that previously we found every end point d went to some interior point \( \tilde {d} \)
  • Look at the orbit of \( \tilde {d} \). Clearly it accumulates at some end point \( p \in E \)
  • There are finitely many d (hence \( \tilde {d} \) ). Thus for these we may arrange \( (x_p , y_p ), p \in E \) such that orbit of d intersects the union of \( (x_p , y_p ), p \in E \)
• After changing \(w_d \), we may assume that every \( \tilde {d} \) belongs to some \( (x_p , y_p ), p \in E \).
• Note that every \( \overset{o}{\mathcal {L}}\) orbit in int D meets some \( (x_p , y_p ) \) infinitely often (why?)
• For \( p \in E \), \(C_p \) = circle obtained by identifying the endpoints of \( [x_p , y_p] \).
• C = disjoint union of these circles.

Goal: Construct a non-nesting system of maps \( \mathcal{M} \) on C with every orbit infinite, in such a way that an \( \mathcal{M} \) -invariant measure on C provides an \( \mathcal{L} \) invariant measure on D

• Construct a finite family \( \mathcal {P} \) of homeomorphisms \( \lambda : U \to V \) between open intervals \( U \subset int D \) and intervals \( V \subset C \). It will have three types of maps:
  1. For each \( p \in E \), include the natural map \( \pi_p : (x_p , y_p) \to C_p \)
  2. Then for each p, we consider the points \( (x_p , y_p ) \) and their common image \( z_p \in C \). Restrict \( \tau_p \) to a homeomorphism between a small neighborhood \( X_p \) of \(x_p\) and a neighborhood \( Y_p \) of \( y_p \). Since \( \tau_p \) preserves orientation, both these neighborhoods are naturally homeomorphic to a neighborhood \( Z_p \) of \( z_p \) in \(C_p \). We include the maps \( X_p \to Z_p \) and \( Y_p \to Z_p \)
  3. The ranges of the maps constructed so far cover C. If domains cover int D then we may stop. Otherwise, we need a third type of maps. Let \( x \in int D \). Some \( \overset{o}{\mathcal {L}}\) word \( \alpha_x \) sends a neighborhood \(U_x \) of x to an open interval \( V_x \) contained in some \( (x_p , y_p) \). The word \( \alpha_x \) may be chosen to be constant near the end point of D. But compactness (?), we deduce that int D may be covered by finitely many \( U_x \). We include the corresponding finite set of maps from \( U_x \to \pi_p (V_x) \subset C_p \)
• This defines the family \( \mathcal {P} \). We now use it to carry \( \overset{o}{\mathcal {L}}\) over to a system \( \mathcal {M} \) on C.
  • Let \( \gamma_1 : U_1 \to V_1 \) and \( \gamma_2 : U_2 \to V_2 \) be two elements of \( \mathcal{P} \). If \( U_1 \cap U_2 \neq \phi \), then \( \gamma_2 \gamma_1^{-1} \) is a homeomorphism between two subintervals of C. We include it in \( \mathcal{M} \).
  • Similarly, for \( \theta \in \overset{o}{\mathcal {L}}\) , we include \( \gamma_2 \theta \gamma_1^{-1} \) is its domain in non-empty.
  • \( \mathcal {M} \) = set of all maps thus obtained, for all possible choices of \( \gamma_1, \gamma_2 \in \mathcal {P} \) and \( \theta \in \overset{o}{\mathcal {L}}\) .
  • It is an open system of maps in C.
  • The construction of \( \mathcal {P} \) and \( \mathcal{M} \) was done in such a way that the following properties hold:
    • Given \( \gamma_1, \gamma_2 \in \mathcal {P} \) and an \( \overset{o}{\mathcal {L}}\) word w, given x in the fomain of \( \gamma_2 w \gamma_1^-1 \), there is an \( \mathcal {M} \) word equal to \( \gamma_2 w \gamma_1^{-1} \) on the neighborhood of x.
    • Given an \( \mathcal {M} \) word w, and y in the domain of \( \gamma_2^{-1} w \gamma_1 \) some \( \overset{o}{\mathcal {L}}\) word coincides with \( \gamma_2^{-1} w \gamma_1 \) near y.

• Every \( \mathcal {M} \) orbit is infinite, because every \( \overset{o}{\mathcal {L}}\) orbit in int D meets some \( (x_p , y_p) \) infinitely often.
• Also note that \( \mathcal{M} \) is non nesting because every orientation preserving \( \mathcal{M} \) word that has a fixed point is a restriction of the identity (why?).
• Furthermore, any \( \mathcal {M} \) invariant probability measure with no atom lifts to an \( \overset{o}{\mathcal {L}}\) invariant measure on D.
• The existence of the words \( w_d \) (which depends on the non nesting of \( \mathcal {L} \) implies that this measure has finite total mass, hence extends to \( \mathcal{L} \) invariant measure on D.

We have now reduced our problem to finding an \( \mathcal{M} \) invariant measure with no atom (assuming that every minimal set of \( \overset{o}{\mathcal {L}}\) is infinite )

From M to N

• Let \( F \subset C \) be a minimal set of \( \mathcal {M} \).
• It is infinite and has no isolated point.
• For each component \( C_p \) of C that meets F, we choose a collapsing map \( \rho_p : C_p \to C_p’\) where \( C_p’ \) is another circle and \( \rho_p \) sends each component of \( C_p \backslash (F \cap C_p ) \) to a point.
• Let C’ be the union of circles \( C_p’\)
• Given an element \( \phi : A \to B \) of \( \mathcal {M} \) whose domain meets F, we consider images A’, B’ of A, B in C’ and the natural homeomorphism \( \phi’ \) between interiors of A’ and B’ (note that A’ , B’ are non degenerate intervals, but they need not be open).
• \( \mathcal{N} \) = collection of these \( \phi’\) . It is an open system of maps on C’.
• \( \mathcal {N} \) is non -nesting and every regular orbit is dense, because every \( \mathcal{M} \) orbit contained in F is dense in F (why?) but there may be finite singular orbits.

From N to O

• Choose an interval [x, y] disjoint from all finite singular orbits such that some orientation preserving \( \mathcal {N} \) word sends x to y.
• Perform the same operation as in (L to M) so as to obtain a system \( \mathcal{O} \) on the circle obtained by identifying the endpoints of [x, y]
• This last system is open, non-nesting with every orbit dense.
• By Sacksteder’s theorem it admits an invariant measure.
• This measure first lifts to an \( \mathcal {N} \) invariant measure, then to the required \( \mathcal {M} \) invariant measure.
• This completes the proof when every minimal set of the original system \( \overset{o}{\mathcal {L}}\) is finite.
• If \( \overset{o}{\mathcal {L}}\) has an infinite minimal set F, we first collapse to a point every component of D \F (as in M to N) .
• We obtain a system \( \mathcal {N} \) on a multi-interval with every regular orbit dense and we deal with it as before.