Local Trivializations of Suspended Minimal Cantor Systems and the Stable Orbit-Breaking Subalgebra

Abstract

It is introduced an analogue of the orbit-breaking subalgebra for the case of free flows on locally compact metric spaces, which has a natural approximate structure in terms of a fixed point and any nested sequence of central slices around this point. It is shown that in the case of minimal flows admitting a compact Cantor central slice, the resulting $C^*$-algebra is the stabilization of the Putnam orbit-breaking subalgebra associated to the induced homeomorphism on the central slice. This construction provides an alternative characterization (up to stabilization) of the orbit-breaking subalgebra introduced by Putnam for minimal homeomorphisms of Cantor spaces in terms of suspension flows associated to such dynamical systems.

Appendix

This appendix contains basic facts about flows on metric spaces that are used throughout the paper.

Lemma 4.1

Let X be a locally compact metric space and $\phi : {\mathbb {R}}\curvearrowright X$ a flow. Denote by ${\mathcal {P}}_c(X)$ the set of compact subsets of X, endowed with the Hausdorff distance $d_H$. If $t_1<t_2$ and $Y: [t_1,t_2] \rightarrow {\mathcal {P}}_c(X)$, $t \mapsto Y_t$ is continuous, then

$$\begin{aligned} \bigcup _{t \in [t_1,t_2]} (t, Y_t) \subset [t_1,t_2] \times X \end{aligned}$$

is closed.

Proof

Let $(r,x) \in ([t_1,t_2] \times X )\backslash \bigcup _{t \in [t_1,t_2]} (t, Y_t)$. Let ${\bar{d}}$ be the metric on $[t_1,t_2] \times X$ given by ${\bar{d}} ((t_1,x_1), (t_2,x_2))= |t_1-t_2| + d(x_1,x_2)$. This metric generates the topology of $[t_1,t_2] \times X$.

Since $Y_r$ is closed, there is $\epsilon >0$ such that

$$\begin{aligned} \inf _{y_r \in Y_r} d(x, y_r)>\epsilon \end{aligned}$$

(5)

It follows from the hypothesis that there is $\epsilon /8 \ge \delta (\{ Y_t\})=\delta >0$ such that

$$\begin{aligned} d_H (Y_s, Y_t)< \epsilon /8 \qquad \text{ for } \text{ every } |s-t|< \delta . \end{aligned}$$

In particular, using compactness of the $Y_t$’s, for every $y_s \in Y_s$ there is ${\bar{y}}_t \in Y_t$ such that

$$\begin{aligned} d( y_s, {\bar{y}}_t) = \inf _{y_t \in Y_t} d( y_s, y_t ) \le \sup _{y'_s \in Y_s} \inf _{y_t \in Y_t} d(y'_s, y_t) \le d_H (Y_s, Y_t) < \epsilon /8 \end{aligned}$$

(6)

for every $|s-t| < \delta$.

Let $\delta ' ((r,x)) = \delta ' >0$ be such that

$$\begin{aligned} {\bar{d}} ((s,x'), (r,x))< \epsilon /4 \qquad \text{ for } \text{ every } |s-r| < \delta ', \; x' \in B_{\delta '} (x). \end{aligned}$$

Let $y_s \in Y_s$, $0<|s-r| < \min \{ \delta , \delta '\}$ and $x' \in B_{\delta '} (x)$. Let ${\bar{y}}_r$ be given as in (6). Then

$$\begin{aligned} \begin{aligned} \epsilon&\overset{(5)}{<} {\bar{d}} ((r,x), (r,{\bar{y}}_r)) \le {\bar{d}} ((r,x), (s,x')) + {\bar{d}} ((s,x'), (s, y_s)) + {\bar{d}} ((s, y_s), (r,{\bar{y}}_r))\\&\overset{(6)}{<}\epsilon /4 + {\bar{d}} ((s,x'), (s, y_s)) +2( \epsilon /8) \end{aligned} \end{aligned}$$

Hence $\inf _{B_{\delta '} (x)}\{{\bar{d}}((s,x'), (s,y_s))\} > \epsilon /2$ for every $y_s \in Y_s$, with $|s-t| < \min \{ \delta , \delta '\}$ and so there is an open neighborhood of (r, x) that is not contained in $\bigcup _{t \in [t_1,t_2]} (t, Y_t)$, proving that $\bigcup _{t \in [t_1,t_2]} (t, Y_t)$ is closed. $\square$

Lemma 4.2

Let X be a locally compact metric space and $\phi : {\mathbb {R}}\curvearrowright X$ a flow. Let $Y \subset X$ compact, $\epsilon >0$ and $f \in C_0(X)$ be such that $|f||_Y \le \epsilon$. There is $\delta >0$ such that $|f(x)| < 2\epsilon$ for every $x \in \phi _{[-\delta , +\delta ]}(Y)$.

Proof

If the claim does not hold, there are sequences $\{y_n\} \subset Y$ and $t_n \rightarrow 0$ such that $f(\phi _{t_n} (y_n)) \ge 2\epsilon$ for every $n \in {\mathbb {N}}$. By compactness we can replace $\{y_n\}$ with a convergent subsequence $y_n \rightarrow {\bar{y}} \in Y$. Then, by continuity of the flow, ${\bar{y}} = \lim \phi _{t_n}(y_n)$ and $f({\bar{y}}) \ge 2\epsilon$, which is a contradiction. $\square$

Lemma 4.3

Let X be a locally compact metric space and $\phi : {\mathbb {R}}\curvearrowright X$ a flow. For every $x \in X$, $t \in {\mathbb {R}}$, $\epsilon >0$, there is $\delta >0$ such that $\phi _s ({\mathcal {B}}_{\delta } (x)) \subset {\mathcal {B}}_\epsilon (\phi _s (x))$ for every $|s| \le |t|$.

Proof

Suppose there is $\epsilon >0$ for which the claim does not hold. Then there are sequences $\delta _n \rightarrow 0$, $y_n \in {\mathcal {B}}_{\delta _n} (x)$, $s_n \in [-|t|, + |t|]$ such that $d (\phi _{s_n} (y_n), \phi _{s_n} (x)) \ge \epsilon$ for every $n \in {\mathbb {N}}$. Passing to a subsequence we can suppose $s_n \rightarrow {\bar{s}} \in [-|t|, + |t|]$. By continuity of the flow $\epsilon \le \lim d(\phi _{s_n} (y_n), \phi _{s_n} (x)) = d (\phi _{{\bar{s}}} (x), \phi _{{\bar{s}}} (x))=0$, a contradiction. $\square$

Lemma 4.4

Let X be a locally compact metric space and $\phi : {\mathbb {R}}\curvearrowright X$ be a flow. Suppose there are $x\in X$, $C \subset X$ closed, $t \ge 0$ such that $C \cap \phi _{[0,t]} (x) = \emptyset$. Then there are non-empty open sets U containing x and V containing C such that $V \cap \phi _{[0,t]} (U ) = \emptyset$.

Proof

Suppose this is not the case. Then for every choice of U containing x and V containing C there is $0 \le s \le t$ such that $\phi _s (U) \cap V \ne \emptyset$. In particular, we can take sequences of open sets $U^n \subset B_{1/n} (x)$ around x, $V_n (C)$ with $\cap _{n \in {\mathbb {N}}} V_n (C) = C$ and $\{s_n \} \subset [0,t]$ such that $\phi _{s_n} (U^n) \cap V_n \ne \emptyset$ for every $n \in {\mathbb {N}}$. Hence $U_n \cap \phi _{-s_n} (V_n) \ne \emptyset$ for every $n \in {\mathbb {N}}$, $\cap _n (U_n \cap \phi _{-s_n} (V_n)) = \{ x\}$; thus $\phi _{s_n} (x) \in V_n$ for every $n \in {\mathbb {N}}$. Passing to a subsequence we find $s \in [0,t]$ such that $\phi _s (x) \in C$, a contradiction. $\square$

Lemma 4.5

Let X be a locally compact metric space and $\phi : {\mathbb {R}}\curvearrowright X$ a free flow. Suppose there is a flowbox B with non-empty interior admitting a central slice C which is a Cantor space and let $C' \subset C \cap B^\circ$ clopen. Then there is $r >0$ such that $\phi (C' \times (-r,+r))$ is open in X.

Proof

Note that if $\phi (C' \times (-r,+r))$ is contained in $B^\circ$, then it is open in X, being the image of an open set under the map $\phi$. Hence it is enough to show that there is $r >0$ such that $\phi (C' \times (-r,+r)) \subset B^\circ$. Suppose this is not the case. Then there are sequences $\{y_n\} \subset C'$ and $r_n \rightarrow 0$ such that $\phi (y_n, r_n) \notin B^\circ$ for every $n \in {\mathbb {N}}$. By compactness of $C'$ we can suppose that $y_n \rightarrow {\bar{y}} \in C'$. Hence $\phi (y_n, r_n) \rightarrow \phi ({\bar{y}},0) \in C' \subset B^\circ$, contradicting the fact that $\phi (y_n, r_n) \notin B^\circ$ for every $n \in {\mathbb {N}}$. $\square$