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 torusThe 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 $H([x,s])=e^{2\pi is}$ obeys $H({\Phi }_t([x,s]))=e^{2\pi it}H([x,s])$. 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.