Slow Entropy of Some Parabolic Flows

Abstract

We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (i.e., slow entropy) can be computed directly from the Jordan block structure of the adjoint representation. Moreover using uniform polynomial shearing we are able to show that the metric orbit growth (i.e., slow entropy) coincides with the topological one for quasi-unipotent flows (this also applies to the non-compact case). Our results also apply to sequence entropy. We establish criterion for a system to have trivial topological complexity and give some examples in which the measure-theoretic and topological complexities do not coincide for uniquely ergodic systems, violating the intuition of the classical variational principle.

Appendix: Failure of the Variational Principle

We now show the failure of the variational principal in a general setting. We do so even in the smooth category. We first show some certain characterizations of transformations with zero entropy at every scale a.

1.1 Characterization of Kronecker systems

We recall the following result from [Fe]:

Proposition A.1

[Fe]. A measure-preserving transformation or flow f has $h_{\mu ,a_\chi }(f) = 0$ with respect to every family of scales $a_\chi$ if and only if f is measurably conjugate to an action by translations on a compact abelian group

We prove an analogue in the topological category, namely:

Proposition A.2

A minimal homeomorphism $f : X \rightarrow X$ of a compact metric space has $h_{{\text {top}},a_\chi }(f) = 0$ for every family of scales $a_\chi$ if and only if f is topologically conjugate to a translation on a compact abelian group.

We use the following folklore characterization:

Lemma A.3

Let $f : X \rightarrow X$ be a transitive homeomorphism or a transitive flow of a compact metric space. Then f is topologically conjugate to translations on a compact abelian group if and only if $\left\{ f^t \right\} _{t\in {\mathbb {R}}}$ is uniformly equicontinuous.

Proof of Proposition A.2

First, we show that if f is topologically conjugate to a translation on a compact abelian group, it has zero topological slow entropy at all scales. If f is topologically conjugate to a translation of a compact abelian group, we are free to use the bi-invariant metric for which f is an isometry. Hence, the Bowen balls are equal to the standard balls for arbitrary n. In particular, $N_{f,X}(\varepsilon ,T)$ is independent of T for any $\varepsilon$. Since a(T) must tend to $\infty$, we get the result.

For the converse, first fix $\varepsilon >0$. That is, we show that if we have zero entropy at all scales, then f is topologically conjugate to a translation on a compact abelian group. Since f is assumed to have 0 entropy at all scales, $N_{f,X}(\varepsilon ,T) \le N_0$ for some fixed $N_0 = N_0(\varepsilon )$ and all $T>0$. Let $Y_t(\varepsilon ) \subset X^{N_0}$ be the set of $N_0$-tuples such that $(x_i) \in Y_t$ if and only if $\bigcup _i B_f^t(x_i,\varepsilon ) = X$. Then $Y_t(\varepsilon )$ is nonempty for every t and $Y_t(\varepsilon ) \subset Y_s(\varepsilon )$ for $s < t$, so $\bigcap _{t >0} \overline{Y_t(\varepsilon /2)} \not = \emptyset$.

We claim that $\overline{Y_t(\varepsilon /2)} \subset Y_t(\varepsilon )$. Indeed, fix t and suppose that $(x_i^k) \in Y_t(\varepsilon /2)$ is a sequence of $N_0$-tuples converging as $k\rightarrow \infty$ to a $N_0$ tuple $(x_i)$. The relation $(x_i^k) \in Y_t(\varepsilon /2)$ is equivalent to the property that if $y \in X$, there exists some i such that $d(f^{t'}(x_i^k),f^{t'}(y)) < \varepsilon /2$ for every $0 \le t' \le t$. Since d and $f^{t'}$ are continuous for each $t'$, $d(f^{t'}(x_i), f^{t'}(y)) \le \varepsilon /2 < \varepsilon$. This proves the claim.

We may thus find an $N_0$-tuple $(x_i)$ so that $\bigcup _i \overline{B_f^t(x_i,\varepsilon )} = X$ for every t. Since the sets $B_f^t(x_i,\varepsilon )$ are nested with respect to t, and each point must lie in such a neighborhood for every t, the sets $B_f^\infty (x_i,\varepsilon ) = \bigcap _{t > 0} \overline{B_f^t(x_i,\varepsilon )}$ still cover X. They are closed sets, but since finitely many cover the space X, at least one must have nonempty interior. That is, for some $z \in Z$ and $0< \gamma < \varepsilon /2$, we have $B(z,\gamma ) \subset \overline{B_f^t(x_i,\varepsilon )} \subset B_f^t(x_i,2\varepsilon )$ for every $t > 0$.

Fix $z\in X$. Since f is assumed to be minimal, for every $\gamma >0$ there exists a $T_\gamma >0$ such that $\bigcup _{t=0}^{T_\gamma }f^{-t}z$ is $\gamma$-dense ($T_\gamma$ depends on z, but since z is an apriori fixed point, we drop it from the notation). Equivalently, $\bigcup _{t= 0}^{T_\gamma } f^{-t}(B(z,\gamma )) = X$. This means that for every $x\in X$ there exists $T_x\in [0,T_\gamma ]$ such that $f^{T_x}(x) \in B(z,\gamma )$. Notice that the family $\{f^t\}_{t\in [0,T_\gamma ]}$ is compact (by compactness of $[0,T_\gamma ]$). Since any compact family of homeomorphisms of a compact space is uniformly equicontinuous, we may choose $\delta >0$ so that $f^t(B(x,\delta )) \subset B(f^t(x),\gamma )$ for $t \le T_\gamma$.

Now, given any $x \in X$, by choice of T, we may find $t \le T$ such that $d(f^t(x),z) < \gamma$. Furthermore, by choice of $\delta$, we conclude that $f^t(B(x,\delta )) \subset B(f^t(x),\gamma ) \subset B(z,2\gamma ) \subset B_f^s(x_i,\varepsilon )$ for every $s > 0$. Then $f^{t+s}(B(x,\delta )) \subset B(f^s(x_i),\varepsilon )$, and if $d(x,y) < \delta$, $d(f^t(x),f^t(y)) < 2\varepsilon$ for every $t \ge 0$. Since $\varepsilon$ was arbitrary, we conclude that $\left\{ f^t \right\}$ is uniformly equicontinuous, and the result. $\quad \square$

1.2 Failure of the variational principle

If $f : X \rightarrow X$ is a homeomorphism of a compact metric space, let ${\mathcal {M}}(f)$ denote the space of invariant measures. Recall that a system is Kronecker if it has pure-point spectrum. The results of the previous section show that if we can find a minimal topological system $f : X \rightarrow X$ such that ${\mathcal {M}}(f)$ is finite dimensional and every $\mu \in {\mathcal {M}}(f)$ is Kronecker,³ but which is not topologically conjugate to a translation on a compact abelian group, then the variational principle for the usual entropy theory will fail for slow entropy. Such systems can be found in non-standard realization theory, and the approximation-by-conjugation method first used by Anosov and Katok in [AK]. We document the conclusion here:

Proposition A.4

Let M be a manifold with a free circle action. Then there exists a uniquely ergodic, volume preserving, $C^\infty$ diffeomorphism $f : M \rightarrow M$ which is measurably conjugate to a translation on a torus ${\mathbb {T}}^d$, $d \ge 1$.

Since a toral translation has entropy zero at all scales and since in the above example we may assume that the diffeomorphism is not uniformly equicontinuous, we obtain from the previous result the following corollary:

Corollary A.5

Let M be a manifold with a free circle action. Then there exists a $C^\infty$ diffeomorphism $f : M \rightarrow M$ and family of scales $\left\{ a_\chi \right\}$ such that:

$$\begin{aligned} \sup _{\mu \in {\mathcal {M}}(f)} h_{\mu ,a_\chi }(f) < h_{{\text {top}},a_\chi }(f). \end{aligned}$$

A similar example can be found by using an example due to Furstenberg [Fur], found in [Kat-Has]:

Proposition A.6

There exists a minimal $C^\infty$ diffeomorphism $f : {\mathbb {T}}^2 \rightarrow {\mathbb {T}}^2$ which is measurably conjugate to $(x,y) \mapsto (x+\alpha ,y)$ for some $\alpha \in S^1$. In particular,

$$\begin{aligned} \sup _{\mu \in {\mathcal {M}}(f)} h_{\mu ,a_\chi }(f) < h_{{\text {top}},a_\chi }(f). \end{aligned}$$

Another interesting class we note are the Sturmian sequences. These are zero entropy closed subshifts on two symbols which are uniquely ergodic, and whose unique measure gives a transformation measurably isomorphic to a circle rotation. They can be obtained in the smooth category by taking the non-wandering set of the $C^1$ Denjoy examples of a nontransitive diffeomorphism. Again, because the only translations on compact abelian groups which are Cantor sets are the odometers, we get the following result:

Proposition A.7

Any Sturmian subshift $\sigma$ preserving an invariant measure $\mu$ satisfies

$$\begin{aligned} 0 = h_{\mu ,a_\chi }(\sigma ) < h_{{\text {top}},a_\chi }(\sigma ), \end{aligned}$$

for some family of scales $\left\{ a_\chi \right\}$.

These examples motivate the following questions:

Question A.8

For which continuous transformations $T : X \rightarrow X$ the variational principle

$$\begin{aligned} \sup _{\mu \in {\mathcal {M}}(T)}h_{\mu ,a_\chi } = h_{{\text {top}},a_\chi }, \end{aligned}$$

holds true for every family of scales $a_\chi$?

Another observation on these examples is the following: for a given Kronecker system, one may find an “ideal” realization which links the measurable orbit growth structure with the topological orbit growth structure. In particular, we ask the following question:

Question A.9

Which measure preserving transformations T of a probability space admit, as a measurable factor, a (unique) topological system (Y, S) such that the pushforward measure has full support in Y and the equality of entropies $h_{\mu , a_\chi }(T)=h_{\text {top}, a_\chi }(S)$ holds at all scales $a_\chi$?