= Suspension flow
A <suspension flow> turns a discrete-time map into continuous motion by moving along an extra time coordinate. For a <diffeomorphism> $f:M\to M$ and constant roof one, form the <mapping torus>
$$
(M\times[0,1])/((x,1)\sim(f(x),0)).
$$
The flow moves vertically at unit speed; each crossing of the top applies $f$ and re-enters at height zero. If $M$ has dimension $d$, the suspension has dimension $d+1$.
Suspending a <hyperbolic toral automorphism> gives an <Anosov flow>. After $k$ roof crossings, its transverse derivative is $A^k$, so the base map's stable and unstable exponential bounds become the transverse flow bounds. A remaining fractional crossing has a uniformly bounded effect on a compact mapping torus. The vertical direction supplies the neutral flow direction. In particular, the two-dimensional <cat map> gives a three-dimensional suspension.
A constant roof also shows why this flow need not be mixing. The well-defined height function $H([x,s])=e^{2\pi i s}$ obeys $H(\Phi_t([x,s]))=e^{2\pi i t}H([x,s])$. Under the product invariant probability measure its mean is zero, but its autocorrelation has modulus one for all $t$. Thus even a mixing base map can produce a suspension with persistent clock correlations.
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