When is the time one map of a suspension flow ergodic?

Question

I'm sure the answer to the following question is well known but I couldn't find the answer I needed.

Let $(\Sigma,\sigma)$ denote the full shift on $k$ symbols and let $\mu$ be an invariant measure for which the system is ergodic. Let $f:\Sigma\to\mathbb R^+$ be a Holder continuous roof function. Let $$\Sigma_f:=\{(\underline a,t):\underline a\in\Sigma, 0\leq t\leq f(\underline a)\} $$ and let $\mu_f$ denote the measure $(\mu\times Leb)|_{\Sigma_f}$. Let $\phi$ denote the usual suspension flow on $\Sigma_F$, it is known that $(\Sigma_f, \phi,\mu_f)$ is ergodic.

Let $T:\Sigma_f\to\Sigma_f$ denote the time one map associated to $\phi$, $T$ preserves measure $\mu_f$. Under what conditions on $f$ is the system $(\Sigma_f, T, \mu_f)$ ergodic?

My guess is that the system is ergodic unless $f$ is cohomologous to a rational constant, but if anyone knows the correct statement and a reference for it that would be great.

Answer 1

Rufus Bowen proved a topological version in [1] for a basic set $X$ of an Axiom A flow $\phi_t:M\to M$. Under your setting, Theorem 3.2 in [1] gives a dichotomy of $X$:

• either $\Phi$ on $X$ is the constant suspension of a basic set of Axiom A map,

• or the (strong) unstable leaf $W^u(p)$ is dense in $X$ for each periodic point $p\in X$.

In the first case, it is clear that the time-1 map $T$ is ergodic if and only if the constant is irrational.

In the second case, the time-1 map $T:X\to X$ is topologically mixing. Under your setting, the system $(T,\mu_f)$ should be ergodic, but I didn't find a short argument.

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[1] R. Bowen. Periodic Orbits for Hyperbolic Flows. Amer. J. Math. 94, No. 1 (1972), 1--30.