K-flow with a finite positive entropy

Question

I have a task to give an example of a $K$-flow with a finite positive entropy. I am studying a book called "Ergodic Theory" by Kornfeld, Sinai, Fomin. I found a statement there that for every $K$-flow $T$ its entropy $h(T)>0$. But i can't come up with an example of a finite one. I know how to count entropy for Bernoulli K-automorphisms but i don't know what to do in a continious case. Thank you for any help

Answer 1

The short answer is: Classical systems have finite entropy. Bernoulli systems have the $K$-property. Geodesic flow on a closed manifold with negative sectional curvature is a classical system that is Bernoulli (by a now classical result by Ornstein and Weiss (see their paper "Geodesic flows are Bernoullian")); hence it's an example of a $K$-flow with finite (and positive) entropy.

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I'd like to offer a more constructive approach for a longer answer.

There are two approaches to the matter of finiteness of entropy: one from the point of view of abstract ergodic theory and the other from the point of view of ergodic theory of topological/smooth dynamical systems. There doesn't seem to be general principles that guarantee finiteness of entropy from the point of view of abstract ergodic theory, but under fairly general conditions finiteness of entropy is guaranteed.

All of the results I'm citing below are by now standard in (smooth) ergodic theory circles; their proofs can be found in the books of Walters, Petersen, Mañé, Brin-Stuck or Katok-Hasselblatt (or Cornfeld-Fomin-Sinai). Also Katok's survey "Fifty Years of Entropy in Dynamics: 1958-2007" (https://www.personal.psu.edu/axk29/pub/E50-aspublished.pdf) and Kawan's survey "Topological Entropy - A Survey" (https://www.christophkawan.de/img/Entropy_v30.pdf) are worth taking a look. I hope what I write below will be useful at least for navigation purposes.

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Let us first consider the case of cascades (i.e. $\mathbb{Z}$-actions); we'll extrapolate to the case of flows (i.e. $\mathbb{R}$-actions) later, as the entropy of a flow is by definition the entropy of its time-$1$ map (as in Def.6 on p. 254 of Cornfeld-Fomin-Sinai).

First we have that topological entropy bounds any metric entropy from above:

Theorem (Goodwyn): Let $X$ be a compact metrizable space, $f:X\to X$ be continuous. Then for any $f$-invariant Borel probability measure $\mu$ on $X$, we have

$$0\leq\operatorname{ent}_\mu(f)\leq \operatorname{topent}(f)\leq\infty.$$

(Of course it is true that topological entropy is approximated my metric entropies arbitrarily well (which statement is the content of the Variational Principle, e.g. as discussed in Relation between topological entropy and metric entropy), but we don't need this to bound entropy from above.)

Thus to bound metric entropy from above it is sufficient to bound topological entropy from above. The most general such upper bound is this:

Theorem: Let $(X,d)$ be a compact metric space. If $f:X\to X$ is Lipschitz, then

$$0\leq\operatorname{topent}(f)\leq \underline{\operatorname{boxdim}}(X,d)\, \max\{0,\log(\operatorname{Lip}_d(f))\}\leq\infty,$$

where $\underline{\operatorname{boxdim}}(X,d)$ is the lower box dimension of $X$ and $\operatorname{Lip}_d(f)=\sup_{x\neq y}\dfrac{d(f(x),f(y))}{d(x,y)}$ is the Lipschitz constant of $f$.

(For a proof of this see p. 7-12 of https://www.math.stonybrook.edu/~jack/DYNOTES/dn7.pdf .)

In particular we have our first fairly general upper bound for entropy:

Corollary (Kushnirenko): Let $M$ is a compact Riemannian $C^1$ manifold. If $f: M\to M$ is $C^1$, then

$$0\leq\operatorname{topent}(f)\leq \dim(M)\, \max\{0,\log(\Vert Tf\Vert)\}<\infty,$$

where $\Vert Tf\Vert=\sup_{x\in M} |T_xf: T_x M\to T_{f(x)}M|$ is the global operator norm of the derivative of $f$.

Kushnirenko's bound is typically referred to by the motto "Classical systems have finite entropy".

Let us end this section with an explicit example:

Example: Let $d\in\mathbb{Z}_{\geq1}$, and take $f: \mathbb{T}^d\to \mathbb{T}^d$ to be a surjective topological group homomorphism of the $d$-torus. In an earlier answer (https://math.stackexchange.com/a/4320757/169085) I've established that $f$ preserves the Haar probability measure on $\mathbb{T}^d$ and that $f$ is represented by a $d\times d$ matrix $A$ with integer entries. If $\mu$ is an anonymous $f$-invariant Borel probability measure on $\mathbb{T}^d$, then

$$0\leq \operatorname{ent}_\mu(f)\leq \operatorname{topent}(f)=\operatorname{ent}_{\operatorname{haar}}(f)=\sum_{\substack{\lambda\in\operatorname{Spec_\mathbb{C}(A)}\\|\lambda|>1}}\log(\lambda)<\infty.$$

Note that for $d=1$ taking the doubling map on the circle recovers $\log(2)$ as the upper bound, which is, as expected, the metric entropy of shift on the sequence space with two symbols (with respect to the product measure).

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Next, let us relate flows to cascades. According to p.254 of Cornfeld-Fomin-Sinai, if $\phi_\bullet:\mathbb{R}\to \operatorname{Aut}(X,\mu)$ is a measurable flow by bimeasurable measure preserving automorphisms of a standard probability space (the measurability of the flow refers to the measurability of the map $\mathbb{R}\times X\to X, (t,x)\mapsto \phi_t(x)$), then its entropy is defined as the entropy of the time-$1$ map of $\phi_\bullet$, i.e.,

$$\operatorname{ent}_\mu(\phi_\bullet):= \operatorname{ent}_\mu(\phi_1).$$

The reason behind this definition is that the entropy of the time-$t$ map of $\phi_\bullet$ relates nicely to the entropy of the time-$1$ map (Theorem 3 on the next page of CFS):

Theorem (Abramov): Let $(X,\mu)$ be a standard probability space, $\phi_\bullet:\mathbb{R}\to \operatorname{Aut}(X,\mu)$ be a measurable measure-preserving flow. Then

$$\forall t\in\mathbb{R}: \operatorname{ent}_\mu(\phi_t)=|t|\, \operatorname{ent}_\mu(\phi_1).$$

(In fact there is also a first-principles reason, apparent in the topological setting, as to why the entropy of a flow ought to be defined as the entropy of its time-$1$ map.)

Finally recall that the suspension construction (= "special flows" in the language of CFS) allows one to start with a cascade and construct a flow from it. More specifically, let $(X,\mu)$ be a standard probability space, $f:(X,\mu)\to (X,\mu)$ be a measure theoretical automorphism, $\alpha:X\to \mathbb{R}_{>0}$ be a measurable function with $\mathbb{E}_\mu(\alpha)=1$. Then $\alpha$ generates a (real-valued additive) measurable cocycle $A$ over $f$ defined by

$$A:\mathbb{Z}\times X\to \mathbb{R},\quad A(1,x)=\alpha(x),\quad A(n+m,x)=A(n,f^m(x))+A(m,x).$$

Explicitly, this last condition means that for $n\in\mathbb{Z}_{\geq1}$:

\begin{align*} A(n,x)&=\alpha\circ f^{n-1}(x)+\alpha\circ f^{n-2}(x)+\cdots+\alpha\circ f(x)+\alpha(x)\\ A(-n,x)&=-\alpha\circ f^{-n}(x)-\alpha\circ f^{-n+1}(x)-\cdots-\alpha\circ f^{-2}(x)-\alpha\circ f^{-1}(x). \end{align*}

Using $A$ we have a diagonal cascade on $\mathbb{R}\times X$ generated by $^{A}f:(t,x)\mapsto (t-\alpha(x),f(x))$ (The time-$n$ map of this cascade is $\left( ^{A}f\right)^n:(t,x)\mapsto (t-A(n,x),f^n(x))$.). Quotienting out the orbits we get the suspension space $\mathbb{R}\otimes_{f,A} X$, which is a standard probability space when endowed with the measure $\operatorname{leb}\otimes_{f,A}\,\mu$. (A straightforward way of seeing how to integrate w/r/t this measure is to note that the hypograph (or more accurately "sinistergraph", based on my conventions) $\{(t,x)\in\mathbb{R}\times X\,|\, 0\leq t\leq\alpha(x)\}$ of $\alpha$ is a fundamental domain for the canonical projection $\mathbb{R}\times X\to \mathbb{R}\otimes_{f,A} X$).

The left addition flow $l_s:(t,x)\mapsto (s+t,x)$ on $\mathbb{R}\times X$ commutes with the canonical projection, hence descends to a measurable measure-preserving flow $\hbar^{f,A}_\bullet$ on the suspension space $(\mathbb{R}\otimes_{f,A}X,\operatorname{leb}\otimes_{f,A}\,\mu)$. The flow $\hbar^{f,A}_\bullet$ is called the suspension (or special flow made out) of the automorphism $f$ and cocycle $A$.

(In CFS as well as other classical ergodic theory books the original space is typically considered to be the first (hence horizontal) component, but in more general settings it seems to me that thinking of the original space as the second (hence vertical) component is more beneficial; hence the discrepancy between my notation and what's written in CFS. In particular "$\hbar$" denotes a horizontal flow here.)

Let us suppress $A$ if it is generated by $\alpha:x\mapsto 1$. Note that in the case of $\alpha=1$, if $n$ is an integer, then $\hbar^f_n[0,x]=[n+0,x]=[0+n,x]=[0,f^{-n}(x)]$, thus the subset $X_0=\{[0,x]\,|\, x\in X\}$ of the suspension space is invariant under the time-$1$ map $\hbar^f_1$ of the suspension flow, and the systems $f:(X,\mu) \to (X,\mu)$ and $\hbar^f_1:(X_0,\mu_0)\to (X_0,\mu_0)$ are measure theoretically isomorphic. Here $\mu_0$ is the conditional measure of $\operatorname{leb}\otimes_{f,A}\,\mu$ along the projection $\mathbb{R}\otimes_{f,A}X\to \mathbb{T}$ to the first coordinate, supported on $X_0$. Explicitly, this means that $\mu_0=\delta_0\times \mu$.

(In the general $\alpha$ case there is a similar statement that I'll leave to you to formulate.)

To sum up, the suspension construction not only produces a flow out of a cascade; we also have automatically that the time-$1$ map of the suspension flow is conjugate to the original automorphism (As discussed in CFS, a theorem by Ambrose and Kakutani establishes that all measurable, measure preserving flows (with no fixed points) is in fact the suspension flow of an automorphism and a cocycle, but we won't need this to construct examples.). Since metric entropy is an invariant of conjugacy, we have the following:

Corollary (Abramov): Let $(X,\mu)$ be a standard probability space, $f:(X,\mu)\to (X,\mu)$ be an automorphism. Then

$$\forall t\in\mathbb{R}: \operatorname{ent}_{\operatorname{leb}\otimes_f\mu}(\hbar^f_t)=|t|\, \operatorname{ent}_\mu(f).$$

In particular, a suspension flow over an automorphism $f$ has finite positive entropy if and only if the automorphism $f$ has finite positive entropy.

(Actually, originally the Abramov Theorem above was the corollary of the Abramov Corollary; I'm following CFS's order here. Accordingly, the Abramov Corollary is true for an arbitrary cocycle $A$.)

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Finally let us recall the definition of the $K$-property for measurable cascades and flows. Note that heuristically the $K$-property captures the Kolmogorov zero-one law from probability theory.

Definition: Let $(X,\mathcal{B}(X),\mu)$ be a standard probability space, $f:(X,\mathcal{B}(X),\mu)\to (X,\mathcal{B}(X),\mu)$ be a measure theoretical automorphism. Then $f$ is said to have the $K$-property if there is a sub-$\sigma$-algebra $\mathcal{K}$ of $\mathcal{B}(X)$ such that

1. $\mathcal{K}$ is contained in $f(\mathcal{K})=\{f(K)\,|\, K\in\mathcal{K}\}$,
2. $\mathcal{B}(X)$ is the smallest sub-$\sigma$-algebra of itself containing $f^n(\mathcal{K})$ for any $n\in\mathbb{Z}$,
3. The largest sub-$\sigma$-algebra of $\mathcal{B}$ contained in $f^n(\mathcal{K})$ for any $n\in\mathbb{Z}$ is that $\sigma$-algebra which consists of $\mu$-negligible or $\mu$-conegligible measurable subsets.

Definition: Let $(X,\mathcal{B}(X),\mu)$ be a standard probability space, $\phi_\bullet:\mathbb{R}\to\operatorname{Aut}(X,\mathcal{B}(X),\mu)$ be a measurable measure-preserving flow. Then $\phi_\bullet$ is said to have the $K$-property if there is a sub-$\sigma$-algebra $\mathcal{K}$ of $\mathcal{B}(X)$ such that

1. $\mathcal{K}$ is contained in $\phi_t(\mathcal{K})$ for any $t\in\mathbb{R}_{>0}$,
2. $\mathcal{B}(X)$ is the smallest sub-$\sigma$-algebra of itself containing $\phi_t(\mathcal{K})$ for any $t\in\mathbb{R}$,
3. The largest sub-$\sigma$-algebra of $\mathcal{B}$ contained in $\phi_t(\mathcal{K})$ for any $t\in\mathbb{R}$ is that $\sigma$-algebra which consists of $\mu$-negligible or $\mu$-conegligible measurable subsets.

The sentence right after this definition in CFS (p. 280) together with Theorem 2 in CFS (p. 288) relates the $K$-property for a flow and the $K$-property for its time-$t$ maps:

Theorem: Let $(X,\mathcal{B}(X),\mu)$ be a standard probability space, $\phi_\bullet:\mathbb{R}\to\operatorname{Aut}(X,\mathcal{B}(X),\mu)$ be a measurable measure-preserving flow. Then the following are equivalent:

1. The flow $\phi_\bullet$ has the $K$-property.
2. For any $t\in\mathbb{R}\setminus0$: the automorphism $\phi_t$ has the $K$-property.
3. For some $t^\ast\in \mathbb{R}\setminus0$: the automorphism $\phi_{t^\ast}$ has the $K$-property.

[This is wrong; see the discussion at https://mathoverflow.net/q/412526/66883. More precisely, it is not true that the suspension of a $K$-automorphism with a constant cocycle is a $K$-flow. I will correct this part. I apologize for the mistake.]

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Assembling everything, we have what we need: If $f:(X,\mu)\to (X,\mu)$ is a measure theoretical automorphism that has the $K$-property, and if $\operatorname{ent}_\mu(f)<\infty$, then the suspension flow $\hbar^f_\bullet:\mathbb{R}\to \operatorname{Aut}(\mathbb{R}\otimes_f X, \operatorname{leb}\otimes_f \mu)$ is a measurable measure-preserving flow that has the $K$-property whose metric entropy is finite.

(As you have mentioned $K$-property guarantees positivity of entropy.)

You seem to have no issues with automorphisms with $K$-property, so I'm leaving it to you to generate examples.

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As a final note, observe that the first example I've mentioned is not covered by the long answer I gave. Indeed, giving examples of flows with $K$-property is one thing, verifying whether a flow has the $K$-property is another, especially if the flow is coming from differential geometry.

As a further exercise, it might be interesting to think about flows that have the $K$-property but are not Bernoulli.