Nonautonomous and Stochastic Dynamical Systems

Research Article . DOI: 10.2478/msds-2013-0003 . NSDS . 2014 . 26-52

VERS ITA

Metric Entropy of Nonautonomous Dynamical Systems

Abstract

We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (X„,y„) of probability spaces and a sequence of measurable maps f„ : X„ ^ X„+ with f„^„ = ¡j„+1 . This notion generalizes the classical concept of metric entropy established by Kol-mogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz, and Snoha. Moreover, it shares several properties with the classical notion of metric entropy. In particular, invariance with respect to appropriately defined isomorphisms, a power rule, and a Rokhlin-type inequality are proved.

Keywords

Nonautonomous dynamical systems • topological entropy • metric entropy • variational principle •

MSC: 37B55, 37A05, 37A35 © Versita sp. z o.o.

1. Introduction

In the theory of dynamical systems, entropy Is an Invariant which measures the exponential complexity of the orbit structure of a system. Undoubtedly, the most important notions of entropy are metric entropy for measure-theoretic dynamical systems, sometimes also named Kolmogorov-Sinai entropy by its inventors, and topological entropy for topological systems (cf. Kolmogorov [12], Sinai [25], and Adler et al. [1]). There exists a huge variety of modifications and generalizations of these two basic notions. However, most of these only apply to systems which are governed by time-invariant dynamical laws, so-called autonomous dynamical systems. In the literature, one basically finds two exceptions. In the theory of random dynamical systems, which are nonautonomous dynamical systems described by measurable skew-products, both notions of entropy, metric and topological, have been defined and extensively studied (see, e.g., [3, 7, 17, 18, 27]). In particular, the classical variational principle, which relates the two notions of entropy to each other, has been adapted to their random versions by Bogenschutz [3]. The second exception is the quantity introduced in Kolyada and Snoha [13], the topological entropy of a nonautonomous system given as a discrete-time deterministic process on a compact topological space. The theory founded in [13] has been further developed in [5, 9, 10, 14, 20, 22, 26, 28, 29] by several authors. In some of these articles, the definition of entropy has been extended, in particular to continuous-time systems, to systems with noncompact state space, systems with time-dependent state space, and to local processes. Besides that, there have been other independent approaches (see, e.g., [21, 24]), which essentially lead to the same notion. Both of the nonautonomous versions of entropy, random and deterministic, are intimately related to each other but nevertheless, one cannot draw direct conclusions from the well-developed random theory to the deterministic one except for generic statements (saying that something holds for almost every deterministic system in a large class of such systems parametrized by a random parameter).

* E-mail: christoph.kawan@math.uni-augsburg.de

Christoph Kawan1*

1 Institut für Mathematik, Universität Augsburg, 86159 Augsburg, Germany

Received 24-05-2013 Accepted 24-06-2013

The reason why the deterministic nonautonomous theory of entropy Is still quite poor-developed In particular lies In the fact that the notion of metric entropy (together with a variational principle) has not yet successfully been established in that theory. To the best of my knowledge, the only approach in this direction can be found in Zhu et al. [28]. This work shows that one of the obstacles in establishing a reasonable notion of metric entropy which allows for a variational principle lies in the proof of the power rule which relates the entropies of the time-f-maps (the powers of the system) to that of the time-one-map. The aim of this paper is to introduce the notion of metric entropy for nonautonomous measure-theoretic dynamical systems together with a formalism which allows for a power rule and at least the easier part of the variational principle.

We briefly describe the contents of the paper. In Section 2, we recall the notion of topological entropy for a nonautonomous dynamical system as defined in [14] by Kolyada, Misiurewicz, and Snoha. This notion generalizes the one in [13] by replacing the state space X (a compact metric space) by a whole sequence Xn of such spaces. The process is then given by a sequence of continuous maps fn : Xn — Xn+1. As in the classical theory, three equivalent characterizations of entropy are available, via open covers, via spanning sets, or via separated sets. However, one crucial point here is that in the open cover definition, sequences of open covers for the spaces Xn with Lebesgue numbers bounded away from zero have to be considered. In order to prove the power rule for this entropy, the additional assumption that the sequence fn be uniformly equicontinuous is necessary.

In Section 3, the metric entropy is defined. Here the system is given by a sequence fn : Xn — Xn+1 of measurable maps between probability spaces (Xn,^n) such that the sequence ¡jn of measures is preserved in the sense that fnyn = fjn+1. The metric entropy with respect to a sequence of finite measurable partitions of the spaces Xn can be defined in the usual way (with the obvious modifications), and has similar properties as in the autonomous case. Similarly as in the topological situation (the definition of entropy via sequences of covers), one does not get a reasonable quantity by considering all sequences of partitions. One problem is that information about the initial state can be generated merely due to the fact that the partitions in such a sequence become finer very rapidly. Hence, we have to restrict the class of admissible sequences of partitions, which is done in an axiomatic way by requiring some of the properties that are satisfied in the topological setting by the class of all sequences of open covers with Lebesgue numbers bounded away from zero. This leads to the notion of an admissible class which enjoys some nice and natural properties. For instance, in the case of an autonomous measure-preserving system, one can consider the smallest admissible class which contains all constant sequences of partitions, which leads to the classical notion of metric entropy. Several properties of the classical metric entropy carry over to its nonautonomous generalization. In particular, we can establish an analogue of the Rokhlin inequality, invariance under appropriately defined isomorphisms, and a power rule.

In Section 4, we prove for equicontinuous systems the inequality between metric and topological entropy which establishes one part of the variational principle. We adapt the arguments of Misiurewicz's elegant proof from [19] by defining an appropriate admissible class of sequences of partitions which is designed in such a way that Misiurewicz's arguments can be applied to its members. This class depends on the given invariant sequence of measures. In general, it might be very small, so that our variational inequality would not give any meaningful information. For this reason, we establish different stability conditions for invariant sequences of measures which guarantee that the associated Misiurewicz class contains sequences of arbitrarily fine partitions. These stability conditions capture the intuitive idea that the initial measure ¡j1 should not be deformed too much by pushing it forwards by the maps f" = fn o • • • o f1, so that such sequences become an appropriate nonautonomous substitute of invariant measures. In particular, we show that the expanding systems studied in Ott, Stenlund, Young [23] satisfy such a stability condition with respect to smooth initial measures.

2. Preliminaries 2.1. Notation

By a nonautonomous dynamical system (short NDS) we understand a deterministic process (X1^1,f1i^1), where = {Xn}n>1 is a sequence of sets and fn : Xn — Xn+i a sequence of maps. For all integers k, n e N we write

f0 := idxk, f" := fk+n-1) ◦•••◦ fk+i o fk, f-n := (f.

The last notation will only be applied to sets. We do not assume that the maps fn are invertible. The trajectory of a point x e Xi is the sequence {f"^)}"^. By fkx, we denote the sequence {fk, fk+1, fk+2,...} which defines a NDS on

Xk,<x = {Xk, Xk+i, Xk+2,...}.

We consider two categories of systems, metric and topological. In a metric system, the sets Xn are probability spaces and the maps fn are measure-preserving. That is, each Xn is endowed with a c-algebra An and a probability measure ¡j„ such that the maps fn are measurable and fn^n = )jn+1 for all n > 1, where fn^n denotes the push-forward (fn^n)(A) = IJn(f-1 (A)) for all A G An+1. In this case, we call ¡j1= {^n}n>1 an f1TO-invariant sequence. In a topological system, each Xn is a compact metric space and the maps fn are continuous.

If X is a compact topological space and U an open cover of X, we denote by M(U) the minimal cardinality of a finite subcover. If U1,..., Un are open covers of X, we write \f"i=1 U' for their join, i.e., the open cover consisting of all the intersections Uh n Ui2 n ... n Uin with Uj G Uj.

In a metric space (X, q), we denote the open ball centered at x with radius e by B(x, e) or B(x, e; q). We write dlst(x, A) for the distance from a point x to a nonempty set A, i.e., dist(x,A) = infoeA q(x, a). The closure, the interior, and the boundary of a set A will be denoted by clA, int A, and dA, respectively.

Recall that the Lebesgue number of an open cover U of a compact metric space X is defined as the maximal e > 0 such that every e-ball in X is contained in one of the members of U.

2.2. Topological Entropy

In this subsection, we recall the notion of entropy for a topological NDS (X1to, f1 as defined in Kolyada et al. [ ]. As in the classical autonomous theory, three equivalent definitions are available. We denote the metric of Xk by Qk and define on each of the spaces Xk a class of Bowen-metrics by

QkAx, y) := ^max^ Qk+' (fik(x), fik(y)) (n G N).

It is easy to see that Qkn is a metric on Xk which is topologically equivalent to Qk. In order to define the topological entropy of f1TO, we only use the metrics Q1n. A subset E C X1 is called (n, e)-separated if any two distinct points x, y G E satisfy Q1nn(x, y) > e. A set F C X1 (n, e)-spans another set K C X1 if for every x G K there is y G F with Q1nn(x, y) < e. We let rsep(n, e, f1TO) denote the maximal cardinality of an (n, e)-separated subset of X1 and rspan(n, e, f1TO) the minimal cardinality of a set which (n, e)-spans X1, and we define

1 ( ) hsepf,to) := limlimsup - log rSep (n,e,U,

e\° n^rn n 1

hSpan(f1,^) := lim lim sup — log rSpa„ (n,e,f1^) .

e\0 n^tt n

The corresponding limits in e exist, since the quantities rsep(n, e, f1lXI) and rspan(n, e, f1lXI) are monotone (non-increasing) with respect to e, and this property carries over to their exponential growth rates. Hence, the limits can also be replaced by the corresponding suprema over all e > 0. With the same arguments as in the autonomous case, one shows that the numbers hsep(f1ttl) and hspan(f1ttl) actually coincide. We call their common value the topological entropy of The definition of topological entropy via open covers has to be modified a little bit in order to fit to the nonautonomous case. Consider a sequence = {Un} such that Un is an open cover of Xn for each n > 1. The entropy of with respect to the sequence is then defined as

hcov(f1,^;U^J) := lim sup 1 logM I Y f-'U+A .

n \i=0 I

In contrast to the autonomous case, the upper limit cannot be replaced by a limit (see [13] for a counterexample). In order to define the topological entropy of one should not take the supremum of hcov(f1xi;U1iX>) over all sequences of open covers. The problem is that the value of hcov(f1xi;U1iX>) might become arbitrarily large just by the fact that the maximal diameters of the open sets in the covers Un exponentially converge to zero for n to. In this case, information about the initial state can be obtained due to finer and finer measurements even if the system has very regular dynamics. To exclude this, we restrict ourselves to sequences of open covers with Lebesgue numbers bounded away from zero. We denote the family of all these sequences by £,(X1to) and define

hcov(f1,TOi) := sup hcov(f1,TO;U1to). U1

We leave the easy proof that this number coincides with the topological entropy as defined above to the reader. In the rest of the paper, we write htop(f:for the common value of hsep(f:^), hspan(f:^), and hcov(f: ^).

Remark 1.

Note that the value of htop(f1xi) heavily depends on the metrics Qk in contrast to the classical autonomous situation. However, in many relevant examples, as, e.g., systems defined by time-dependent differential equations, all of these metrics come from a single metric on a possibly compact space. So in this case the dependence on the metrics disappears due to a canonical choice.

The topological entropy of an autonomous system given by a map f satisfies the power rule htop(fk) = k • htop(f) for all k > 1. In order to formulate an analogue of this property for NDSs, we have to introduce for every k > 1 the k-th power system of the NDS (Xi,OT, f1ja>). This is the system (X1^, f^), where

Xll := {X("-1)k+l}n>1 , flkL : = {f[n-1)k+l}n>1 ■

In case that the spaces Xn coincide, the following result can be found in [13, Lem. 4.2]. Since the proof for the general case works analogously, we omit it.

Proposition 2.

For every k > 1 it holds that

htop (f1"l) < k • (fi,„) ■

In general, the converse inequality in the above proposition fails to hold (see [13] for a counterexample). However, if we assume that the family {fn} is equicontinuous, equality holds. Equicontinuity in this context means uniform equicontinuity, i.e., for every e > 0 there exists 5 > 0 such that Qn(x,y) < 5 for any x,y e Xn, n e N, implies Qn+:(fn(x), fn(y)) < e. In [13, Lem. 4.4] this is proved for the case when the spaces Xn all coincide, by using the definition via separated sets. Here we present a different proof using the definition via sequences of open covers, since we want to carry over the arguments later to the proof of the power rule for metric entropy.

Lemma 3.

Let e £.(X1xi) and assume that f1x, is equicontinuous. Then for each m > 1 the sequence V1,x, defined by Vn :=Vm =o fn 'Un+i, is an element of £(X1,tt}).

Proof. Let e > 0 be a common lower bound for the Lebesgue numbers of the covers Un. Then, for each n > 1, e is also a lower bound for the Lebesgue number of Vn with respect to the Bowen-metric gn m. This is proved as follows: Let x e Xn and assume that Qn:m(x, y) < e. Then fn(y) is contained in the ball B(f'n(x), e; Qn+i) for i = 0,1, ■ ■ ■, m — 1. Since e is a lower bound of the Lebesgue number of Un+i for all i, we find sets Ui e Un+i such that B(f'n(x),e; Qn+i) C Ui for i = 0,1, ■ ■ ■, m — 1, which implies that

B(x,e; Qn.m) C U0 n f-1(U1) n f-2(U2) n ■■■ n f—(m—1)(Um-1)

e V f—'Un+i = Vn.

It is easy to see that from equicontinuity of it follows that also the family {f'n : n > 1, i = 0,— 1} is equicontinuous. Hence, we can find 5 > 0 such that Qn(x,y) < 5 implies Qn+i(fn(x),fln(y)) < e for all n > 1 and i = 0,— 1. Therefore, every Bowen-ball B(x,e; Qn,m) contains the 5-ball B(x,5; Qn), which shows that 5 is a lower bound for the Lebesgue numbers of the covers Vn. □

Lemma 4.

Let {an}n> be a monotonically increasing sequence of real numbers. Then for every k > 1 it holds that

an ank

lim sup — = lim sup ——.

n — TO n n — TO nk

Proof. It suffices to prove the inequality "<". To this end, consider an arbitrary sequence {ni}i>1 of positive integers converging to oo. For every i > 1 there is an mi G N0 with mik < ni < (mi + 1)k, and mi —> oo. This implies

niani < mika(mi+1)k.

It follows that

,. a<m.+Vlk mi + 1 .. om.k

lim sup , = lim sup--———— = lim sup —l—.

i^ mik i^ mi (mi + 1)k i^ mik

Hence, we conclude that

Um sup .....

i — TO ni i — TO mik

which yields the desired inequality. □

■ . Qni . .. Qmk . .. Qmk

lim sup —- < lim sup —i— < lim sup ——, ni mik m^m mk

Proposition 5.

If the sequence f1TO is equicontinuous, then

htop (C) = k • ht0p (f^) for all k > 1.

Proof. It suffices to prove the inequality ">". To this end, let G C.(X1toi). Define a sequence Vi,TO = {Vn} of open covers for X^ as follows:

Vn := U(n-1)k+1 V f--1)k+1U(n-1)k+2 V ... V f(n(k-1+1Unk k-1

= V f(n- 1)k+1 U(n-1)k+1+j. j=0

Then we find

hcov (/11; Vi) = lim sup - logM ( \ /-¡k V+

\ ' n^ n \ = I

- / n-i k-i ^

= lim sup - logM (V f-ik V f-k+iUk+i+i n \i=0 i=0 )

n-1 k-1

■-('k+i)U I 1 U(ik+i)+1

- / ' 1 = lim sup - logM ( V V /1

n \i=0 i=0 /

1 ¡n- \

= k • lim sup — log M \J /1 ¡U+1

nnk \ ¡=0 I

= k • hcov (/1,i; U1i) .

To obtain the last equality we used Lemma 4. By Lemma 3, V1,m 6 £(XH), which implies

htop (/1,1) > hcov (/1,1 Vn) = k • hcov (/1,1;U11) .

Since this holds for every U1l 6 C,(X1l), the desired inequality follows. □

Remark 6.

Next to the classical notion of entropy for continuous maps on compact spaces, the notion of topological entropy Introduced above generalizes several other concepts of entropy. Here are three examples:

(i) Topological entropy for uniformly continuous maps on noncompact metric spaces (cf. Bowen [4]): Consider a uniformly continuous map f : X —> X on a metric space X. The topological entropy of f is defined by

htop(f) ■= sup Um Um sup — log rspan(n, e, K),

KcX e\° n^œ n

where the supremum runs over all compact sets K C X and rspan(n, e, K) is the minimal cardinality of a set which (n, e)-spans K. Alternatively, one can take maximal (n, e)-separated subsets of K. If we define for each compact set K C X a NDS f\K,l by

Xn := fn-1(K), fK):= f X : Xn — Xn+i, we see that htop(f) can be written as

htop(f) = sup htop(fiiKl).

(ii) Topological sequence entropy (cf. Goodman [8]): Here the sequence X1oo is constant and the sequence fn is of the form fn = fkn, where f : X — X is a given continuous map and (kn)n>1 an increasing sequence of integers.

(iii) Topological entropy of random dynamical systems (cf. Bogenschutz [3]): Consider a probability space (Q, T, P) with an ergodic invertible transformation & on Q, and a measurable space (X, B). A mapping p : Z x Q x X — X such that (w,x) — p(n, w,x) is T ® B-measurable for all n e Z and p(n + m, w,x) = p(n, dmw, p(m, w,x)) for all n, m e Z and (o>,x) e Q x X is called a random dynamical system on X over &. If X is a compact metric space, B is the Borel a-algebra of X, and the maps p(n, w, •) are homeomorphisms, one speaks of a topological random dynamical system. If U is an open cover of X, one defines for every w e Q

hl0p(p;U) := lim - logN \\l p(i,w)-'U \ . (1)

n ^^o n I v I

\ ¡=0 I

From Kingman's subadditive ergodic theorem it follows that this number exists for almost every w e Q and is constant almost everywhere. Then one can take this constant value (for each U) and define the topological entropy of the random dynamical system by taking the supremum over all open covers U. If we fix one w e Q and consider the number (1), replacing the limit by a lim sup, and then take the supremum over all U, we obtain the topological entropy of the NDS (X1 ^, f1:^) given by Xn := X, fn := p(1, &n-1 w, •).

Remark 7.

It is an interesting fact that not only Bowen's notion of topological entropy for uniformly continuous maps is a special case of the topological entropy for NDSs, but that for an equicontinuous NDS (X1 ^, f1:^) also the converse statement is true: htop(f1^1) can be regarded as the topological entropy of a uniformly continuous map, restricted to a compact noninvariant set. To see this, let X be the disjoint sum of the spaces Xn, i.e.,

X := □ Xn, Q(x,y):.=

Then a uniformly continuous map f : X — X is given by putting f equal to fn on Xn, and we have

\n — m\ If x G Xn, y G Xm, n = m, Qn(x,y) If x, y G Xn.

htopifi,^ = htop(f, Xi).

This observation in particular allows to conclude the power rule from the corresponding power rule for Bowen's entropy. Taking the supremum of htop(f, K) over all compact subsets K of X gives the quantity called the asymptotical topological entropy of f- ^ in [13], defined by limn—» htop(fn,J).

3. Metric Entropy

In this section, we Introduce the metric entropy of a NDS.

3.1. The Entropy with Respect to a Sequence of Partitions

Recall that the entropy of a finite measurable partition V = {P1,..., Pk} of a probability space (X, A, j) is defined by

Hj(V) := - £ j(Pi) log j(P)

where 0 • log 0 := 0, and that it satisfies 0 < Hj (V) < log k. The equality Hj (V) = log k holds iff all members of V have the same measure.

If V and Q are two measurable partitions of X, the joint partition VVQ = {P n Q : P e V, Q e Q} satisfies Hj(V V Q) < Hj(V) + Hj(Q).

Now consider a metric NDS (X1c, f1c, j1cc), where j1cC denotes the sequence of probability measures with fnjn = jn+1. Let V1c = {Vn} be a sequence such that Vn is a finite measurable partition of Xn for every n > 1, and define

Wyc Vi,„) := limsup 1 Hji j/ f-iVM) .

n \i=0 I

We call this number the metric entropy of f1c with respect to V1c. Note that in the autonomous case this definition reduces to the usual definition of metric entropy with respect to a partition. In this case, the lim sup is in fact a limit, which follows from a subadditivity argument. However, in the general case considered here, subadditivity does not necessarily hold. (In [13], one finds a counterexample for the topological case, which can be modified to serve as a counterexample in the metric case, since this system preserves the Lebesgue measure.) For an autonomous system given by a map f with an invariant measure j and a partition V, we also use the common notations hj (f; V) and hj (f) = supV hj (f; V). Several well-known properties of the entropy with respect to a partition carry over to its nonautonomous generalization. In order to formulate these properties, we have to introduce some notation. We say that a sequence V1c of measurable partitions is finer than another such sequence if Vn is finer than Qn for every n > 1 (i.e., every element of Vn is contained in an element of Qn). In this case, we write > Qi,cx>. If and are two sequences of measurable partitions, we define their join V1c V Q1c := {Vn V Qn}n>1. For a sequence V1c and m > 1 we define another sequence V^C(ftc) by

m-1 m-1 m-1

Vf1-iVi+1, Vf2-iV+..... Vf-V+k, ...

i=0 i=0 i=0

Finally, recall the definition of conditional entropy for partitions of a probability space (X, A, j). If A, B e A with j(B) > 0, then j(A|B) := j(A n B)/j(B). If V and Q are two partitions of X, the conditional entropy of V given Q is

Hj(V|Q) := - £ j(Q) £ j(P|Q) log j(P|Q).

QeQ PeV

Some well-known properties of the conditional entropy are summarized in the following proposition (cf., e.g., Katok and Hasselblatt [11]).

Proposition 8.

Let V, Q and K be partitions of X.

(i) Hj(V|Q) = 0 iff Q is finer than V (modulo null sets).

(ii) Hj(V V Q|K) = H^K) + Hj(Q\P V K).

(iii) If K is finer than Q, then Hj(V\K) < Hj(VQ).

(iv) 0 < H„(P\Q) < H„(V).

(v) H„(P\K) < H,(P\Q) + H„(Q\K).

Now we can prove a list of elementary properties of h(f1<x; P1oo) most of which are straightforward generalizations of the corresponding properties of classical metric entropy.

Proposition 9.

For any sequences of finite measurable partitions for X1oo the following assertions hold:

(i) 0 < h(fi^;) < 1imsupn^oa(1/n) ^=1 1og#Pi.

(ii) h(fi,oa;Pyoo V Qi:00) < h(fi,oa;Pyoo) + h(fi:00; Qyoo).

(iii) If Pi.oo h Qi.oo, then h(fi,oa;Pyoo) > h(fi,oa; Qyoo).

(iv) For every k > 1 it holds that

1 Ink-1 \

h{fi,oo;Pi,oo) = limsup — H,1 Y f-'P+i\ ■

n^ nk \ i=0 I

(v) For every m > 1 it holds that h(fii00; V^) = h(fii00; v')m,l(h,^)).

(vi) h(fi,00;Vi,ao) < h(fi,00; Qi,^) + limsupn^(1/n) Hn (VQ).

(vii) h(fk,oa; Vk,oo) = h(floo; Vioo) for all k,l e N.

(viii) Let £ denote the family of all sequences V1i00 of finite measurable partitions for X1sxi with uniformly bounded cardinalities #Vn. Then a metric on £ is given by

dR(Vi:00, Qi:00) := sup H,n (Vn\Qn) + sup H,n (Qn\Vn).

n>1 n>1

Moreover, the function V1i00 ^ h(f1oo;V1i00) is Lipschitz continuous with Lipschitz constant 1 on (£, dR).

Proof. The properties (l)-(Hl) follow very easily from the properties of the entropy of a partition. Property (Iv) Is a consequence of Lemma 4, since the partitions \/n=o f-'Vi+t become finer with increasing n, and hence the sequence n ^ n=0 f-'Vi+1) is monotonically increasing. To show (v), note that for every n > 1 we have the identities

(n-1 \ /n-1 m-1 \

v f-vm (f1i») = h,1 (v f- v f-i1vi+'+1)

i=0 I \i=0 i=0 /

(n-1 m-1 \ I n+m-2 \

v v f-il+i]v+++) = hJ V f-kVk+1 .

i=0 i=o J \ k=0 I

1 I n+m-2 \

h(f1^; (fh<x)) = limsup - H,J v f-kvk+1)

v ' n \ k=0 I

= limsup - H,1 \ v f-k Vk+1 | = h (f1i00; Vy^) ,

n \k=o I

This implies

which concludes the proof of (v). Next, let us prove (v'l): From Proposition 8 (11) It follows that

(n-1 \ I n-1 n—1

Vf—"p<+i] <Hi Vf—iv+ivVf—lQ'+i

1=0 I \ 1=0 1=0

n-1 \ In-1 n-1

= Hj1 V f-Qi+1 + Hm V f1-iVi+1^ f-Qi+1 .

i=0 i=0 i=0

For the last term in this expression we further obtain

(n-1 n-1 \

V f1-iVi+11V f-Qi+1

i=0 i=0

(n-2 n-1 \

V1 V f- V f2-iVi+2| V f-Qi+1 i=0 i=0

(n-1 \ I n-2 n-1 \

V11V f1-iQi+1 + Hj1 f1-1 V f2-iVi+2|V1 V V f-Qi+1.

i=0 i=0 i=0

Now we use Proposition 8 (iii) to see that this sum can be estimated by

(n-2 n-1 \

f-1 V f-V^ V f-Qi+1 .

i=0 i=0

Using the same arguments again, for this expression we find

(n-3 n-1 \

f-1V2 V V f-(i+2)Vi+3| V f--iQ+1 i=0 i=0

= Hj1 (V1 |Q1) + Hj1 | f-1 V2| W f-Qi+1 J

+Hj1 ^ fr(i+2)Vi+3|fr1 V2 V W f1-iQi+1

i=0 i=0

< Hj1 (V1Q1) + Hj1 (f1-1V2|f1-1 Q2)

(n-3 n-1 \

- V f3-iVi+3| V f1-iQi+1 .

i=0 i=0

Using f1j1 = j2, we find that Hj1 (f-1 V2|f-1 Q^ = Hj2(V2|Q2). Going on inductively, we end up with the estimate

n-1 n-1

Hj1 V f1-iV i+1 | V f-Q i+1

i=0 i=0 i=1

Hence, we obtain

M^C V1C < h{f1c; Q1C + lim sup £Hji (Vi |Qi),

n i = 1

which finishes the proof of (vi). Let us prove (vii): For any k e N we find

h(fk,^; Vk,J) = Um sup -H,k Vk V "// f-'n+i !

n^ n \ .=1 j

< lim sup - H„k (Pk) + V №+i

n—™ n [ \ 1=1 )

= lim sup nHn f-1 \\ f-^Pk

n—™ n \ i=1 I

1 /n-2 _. \

= limsup — Hw+1 Y f--+1 P(k+1)+i = h (fk+1.™;Pk+1.™) .

n—™ n \ 1=0 I

Using the elementary property of the entropy of partitions that H(A) > H(B) whenever A is finer than B, the converse inequality is proved by

h(fk.™; Pk,™) = lim sup - H,k Pk V V f-Pk+i)

n—™ n \ i=1 I

> limsup — H„k y fi<iPk+A = h(fk+1.™;Pk+1.™).

nn \i=1 I

This implies (vii). Finally, to prove (viii), note that the assertion that dR is a metric easily follows from the properties of conditional entropy stated in Proposition 8. From item (vi) we conclude the nonautonomous Rokhlin inequality

|h(f1,„;P1,„) - h(f1.™; Q1.™)|

1 n 1 n

< ma^ lim sup (PilQi). lim sup (Qi|Pi)

n—™ n i=1 n—™ n i=1

1 n 1 n

< lim sup £ Hn (PilQi) + lim sup £ Hn (Qi|Pi)

n —n i=1 n —n i=1

< sup H„„ (PnlQn) + sup H,n (QnlPn) .

which finishes the proof of the proposition.

Remark 10.

Note that the equality in item (vii) of the preceding proposition reveals an essential difference between metric and topological entropy of NDSs, since in the topological setting only the inequality

htop(fk,c) < ^top (fk+1 ,c )

holds. A counterexample for the equality is given by a sequence f1c on the unit interval such that f1 is constant and all other fn are equal to the standard tent map. In this case, clearly htop(f1cC) = 0, but htop(fkcC) = log 2 for all k > 2 (see also [13] for a counterexample with htop(fkc) < htop(fk+1c) for all k). Therefore, the notion of asymptotical topological entropy, as defined in [13], has no meaningful analogue for metric systems.

Remark 11.

Item (viii) of the preceding proposition is the nonautonomous analogue of the Rokhlin inequality.

From Proposition 9 (vii) we can conclude a similar result as [13, Thm. A] which asserts that the topological entropy of autonomous systems is commutative in the sense that htop(f o g) = htop(g o f).

Corollary 12.

Consider two probability spaces (X, j) and (Y, v) and measurable maps f : X ^ Y, g : Y ^ X such that fj = v and gv = j. Then j is an invariant measure for g o f, v is an invariant measure for f o g, and it holds that

hv(f o g) = hj(g o f).

Proof. We consider the NDS (X^^.f^^) defined by X^ := {X,Y,X,Y,...} and f^^ : = {f,g,f,g,...}. The corresponding f1oo-invariant sequence of measures is j1oo := {j, v, j, v,...}. Consider a finite partition Q of Y, put V := f-1Q, and V1,00 := {V, Q, V, Q,...}. Using Proposition 9 (iv), we find

h (fi,*, ; Vyao)

= Urn sup;1 H. ( \\ f-lVi+1 )

Zn \ =0 I

1 In-1 n-1 \

= l\m sup — Hj V f^i+i V\/ f-(2l+1)V2l+2

ln \ l=0 l=0 I

1 n-1 n-1

= Urn sup—hJ \/(g ◦ f )-lw\/ (g ◦ f )-lf-1Q

ln \l=0 l=0

= llimsup -HJ\j(g ◦ f)-lr \ =1 h„(g ◦ f;V)

1 n \l=0 I 1

Similarly, we obtain 2h(f2ÊOO ; T2,oo ) = hv (f o g; Q). Hence, from Proposition 9 (vil) we conclude h^ (g o f ; V) = hv (f o g; Q). Since we can choose Q freely, this implies hv (f o g) < h^ (g o f ). Starting with a partition V of X and putting Q := g-1V, we get the converse inequality. □

Remark 13.

In Balibrea, Jiménez López, and Cánovas [2] one finds proofs for the commutativity of metric and topological entropy which are not based on entropy notions for nonautonomous systems. These commutativity properties were first found in Dana and Montrucchio [6]. Later, Kolyada and Snoha [13] rediscovered the commutativity of topological entropy.

We finish this subsection with an example which shows that the entropy h(f1sxi; V1oo) can be arbitrarily large even for a very trivial system.

Example 14.

Let Xi oo, fi oo and j1oo be constant sequences given by Xn = [0,1], fn = Ld[o,i], and jn = A (the standard Lebesgue measure). Consider the family V1oo of partitions given by

pn = {[0,11k"), [1lkn, 21k").....[(kn - 1)lk", 1]}

for a fixed integer k > 2. Then one easily sees that

Hj1 ( V f-VM) = Ha(V") = - log 1 = log k" = " log k,

\ i=0 I i=1

which implies h (f1:00;V1i00) = log k.

From this example one sees that by taking appropriate sequences of partitions, one obtains arbitrarily large values for the entropy of the identity. Here we have the same problem as we had in defining the topological entropy via sequences of open covers. If the resolution becomes finer at exponential speed, one obtains a gain in information which is not due to the dynamics of the system. Hence, in the definition of the metric entropy of f1oo, we have to exclude such sequences.

3.2. Admissible Classes and Metric Entropy of Nonautonomous Systems

To define the entropy of the system (X1oxi,f1oxi ,№,c), we have to choose a sufficiently nice subclass £ from the class of all sequences V1c. Then the entropy can be defined in the usual way by taking the supremum over all V1c £ £. In view of the definition of topological entropy in terms of sequences of open covers and Example 14 it is clear that taking all sequences of partitions is too much. Since there is no direct analogue to Lebesgue numbers for measurable partitions, we introduce suitable classes of sequences of partitions by axioms which reflect some properties of the family £(fi,cx>) defined in Section 2.

Definition 15.

We call a nonempty class £ of sequences of finite measurable partitions for X1c admissible (for f1c00) if it satisfies the following axioms:

(A) For every sequence V1c £ £ there is a bound N > 1 on #Vn, i.e., #Vn < N for all n > 1.

(B) If V1c £ £ and Q1c is a sequence of partitions for X1c with V1c > Q1c, then Q1c £ £.

(C) £ is closed with respect to successive refinements via the action of f1c. That is, if V1oo £ £, then for every m > 1 also PC£ £.

From Axiom (A) it follows that the upper bound in Proposition 9 (i) is always finite. Moreover, by adding sets of measure zero, we can assume that #Vn is constant for every element of £. Axiom (B) says that with every sequence V1c £ £ also the sequences which are coarser than V1oo are contained in £. Axiom (C) will be essential for proving the power rule for metric entropy. It reflects the property of sequences of open covers stated in Lemma 3.

Definition 16.

If £ is an admissible class, we define the metric entropy of f1oo with respect to £ by

h£(f1,c) = h£(f1cC №,c) := sup h(f1c VyC).

V1,c££

Proposition 17.

Given a metric NDS (X1c, f1cC), let £ be the class of all sequences of partitions for X1c which satisfy Axiom (A). Then £ is an admissible class. £ is maximal, i.e., it cannot be extended to a larger admissible class. Therefore, we denote this class by £max or £max(X1,c).

Proof. It is obvious that £ cannot be enlarged without violating Axiom (A). Hence, it suffices to prove that £ satisfies Axioms (B) and (C). If V1c £ £ and Q1c is a sequence of partitions which is coarser than V1c, it follows that #Qn < #Vn for all n > 1, which implies Q1c £ £. Now consider for some V1c £ £ and m > 1 the sequence V1mi(f1,c). We have

m-1 m-1 , « m

< n # [№+•>] = n ^ sup #vn ■

i-n /-n \ e1 I

This implies that £ satisfies Axiom (C). □

The following example shows that £max is in general not a useful admissible class.

Example 18.

We show that h£max(f1cCX1) = oo whenever the maps ft are bi-measurable and the spaces (Xn,^in) are non-atomic. Indeed, for every k > 1 we find a sequence V1cC of partitions with #Vn = k such that h(f1oo; V1cC) = log k, which is constructed as follows. On X1 take a partition V1 consisting of k sets with equal measure 1/k. Then Q2 := f1V1 is a partition of X2 into k sets of equal measure. Partition each element Qt of Q2 into k sets Qt1,..., Qk of equal measure 1/k2. Then define a new partition V2 of X2 consisting of the sets P^ := Q11 U Q21 U ... U Qk1, P| := Q12 U ... U Qk2, ...,

P2 : = Q1k U ... U Qkk. Also V2 is a partition of X2 into k sets of equal measure 1 /k, and V2, Q2 are independent. This implies

h^V vfr1V2) = h„1 (ff 1Q2vf-1V2)

= Hn(Q2 V V2) = Hn(Q2) + Hn(V2) = 2 log k.

Inductively, one can proceed this construction. For i from 1 to some fixed n, assume that Vi is a partition of Xi into k sets of equal measure such that Rn := V1 V ff1V2 V ... V f1 (n 1)Vn consists of kn sets of equal measure. Then consider the partition Qn+1 := f-fR of Xn+1. Let Rn = {R1,..., Rkn } and partition each Ri into k sets of equal measure 1 /kn+1, say Ri = Ri1 U ... U Rik. Define the partition Vn+1 = {P'n+1.....P"k+1} by P/+1 := Rv U ... U Rknj. This gives

H1 (Rn V f-nVn+1) = H„1 (ffnQn+1 V ffnVn+1)

H,n + 1 (Qn + 1 V Vn + 1 ) = H,n+1 (Qn + 1 ) + Hn + 1 (Vn + 1 ) log kn + log k = (n + 1) log k,

which implies h(f1oo; V1oo) = log k for the sequence V1oo = {Vn} obtained by this construction.

As this example shows, we have to consider smaller admissible classes. These are provided by the following proposition whose simple proof will be omitted.

Proposition 19.

Arbitrary unions and nonempty intersections of admissible classes are again admissible classes. In particular, for every nonempty subset T C £max there exists a smallest admissible class £(T) which satisfies T C £(T) C £max (defined as the intersection of all admissible classes containing T). We also call £(T) the admissible class generated by T.

We also have to show that the metric entropy of a NDS generalizes the usual notion of metric entropy for autonomous systems. To this end, we use the following result.

Proposition 20.

Let T be a nonempty subset of £max. Then

H(T) := {Q10 £ £max | 3V1,oo £ T : h(f1,c; Q1,») < h(f1,c; V1,»)} (2)

is an admissible class with T C H(T) C £max. Consequently, £(T) C H(T) and it holds that

h£(T)(f1,oo) = hH(T)(f1,oo) = sup h (f1,oo; V1O .

V1№ £T

Proof. It is obvious that T C H(T) C £max. Clearly, H(T) satisfies Axiom (A). It also satisfies Axiom (B), since any sequence R1oo of partitions coarser than some Q1oo £ H(T) satisfies h(f1oo;R1oo) < h(f1oo; Q1oo) < h(f1oo; V1cC) for some V1oo £ T. With the same reasoning and Proposition 9 (v), we see that H(T) satisfies Axiom (C) and hence is an admissible class. □

The preceding proposition shows not only that there exists a multitude of admissible classes, but also that the metric entropy of f1oo can be equal to any of the numbers h(f1oo; V1oo) by taking the one-point set T := {V1oo} as a generator for an admissible class. The next corollary immediately follows.

Corollary 21.

Assume that the sequences X1oo, f1oo, ¡j1oo are constant, i.e., we have an autonomous system (X, f,^i). Let T be the set of all constant sequences of finite measurable partitions of X. Then h£(T)(f1oo) = h^(f).

3.3. Invariance and Restrictions

In order to be a reasonable quantity, the metric entropy of a system f1sxi should be an Invariant with respect to Isomorphlms. By an isomorphism between sequences (X1oo,^1oo) and (Y1:00,v1:00) of probability spaces we understand a sequence n1oo = {nn} of bi-measurable maps nn : Xn — Yn with nnyn = vn. Such a sequence is an isomorphism between the systems f1<x on X1sxi and g1sxi on Y1<x if additionally for each n > 1 the diagram

X„ --—> X,

"I r+1

•n -* 'n+1

commutes. In this case we also say that the systems f1cxi and g1cxi are conjugate. If the maps nn are only measurable but not necessarily bi-measurable, we say that the systems f1cxi and g1cxi are semiconjugate. The sequence n1sx, is then called a conjugacy or a semiconjugacy from f1oo to g1oo, respectively.

Given two admissible classes £ and T for X1cxi and Y1oo, resp., we also define the notions of £-T-isomorphisms and £-T-(semi)conjugacies via the condition that n1sx, respects £ and T in the sense that

Vyoa = {Vn}n>: eT ^ {n-1(Vn)}n>i e £.

In the case of an isomorphism or a conjugacy, the implication into the other direction must hold as well.

Proposition 22.

Let (X1oo, f1c00, mt.oo) and (Y1c00, gt.oo, "1,00) be metric NDSs with admissible classes £ and T, respectively. Let n1cx be an £-T-semiconjugacy from f1cxi to g1cxi. Then

ht(gi,co) < h£(fi:<x).

Proof. First note that the semiconjugacy identities nn+1 o fn = gn o nn imply g 1 o n1 = ni+1 o f\ for all i. Let V1,oo = {Vn} be a sequence of finite measurable partitions for Y1c00. Fix n e N and Pj. eVi, i = 1,..., n. Then we find

"1 (n g-lPj+1) = m |nr 1 n g-lPji+^ = m (n (g 1o n)-1 Ph+^

= m (f] (n+1 o f1)-1 pk+1J = m | f- f-l<1 PJ+1) .

Define Q1cxi = {Qn} by Qn : = {n-1(P) : P e Vn} for all n > 1. Then Qn is a finite measurable partition of Xn and from the preceding computation we get

Hvi ( V 9—'r+i)= H« ( \/ f—lQi+i

Hence, h(f1^; Q1:00) = h(g1i00; V1ilx). Writing Q1:00 = n-ao(P1,<x>), we find

h.(g1,co) = sup h(g1rao; V1rlx)= sup h(f1:<x; ))

V1eT Vyoo eT

< sup h(f1cc; Q1c) = h£(f1^),

Q1,oo e£

as desired. □

Given a metric NDS (X1(, f1(,№,(), assume that we can decompose each of the spaces Xn as a disjoint union Xn = YnUZn such that fn(Yn) C Yn+1, fn(Zn) C Zn+1 and fJn(Yn) = c for a constant 0 < c < 1. Then let us consider the restrictions of f1( to the sequences Y1( := {Yn} and Z1( := {Zn}, resp., i.e., the systems defined by the maps

Yn — Yn+

hn := fn\zn : Z

It we consider the probability measure vn(A) := ¡j„(A)/c on Yn, It follows that (Y1<<, g1<<, v-i,«) Is also a metric system. If c < 1, we can define a corresponding invariant sequence of probability measures for the system (Zi,« h1<<) as well.

Proposition 23.

Let £ be an admissible class for (X1œ, f1<<) and assume that T1<< G £ implies {Vn V {Yn, Zn}} G £. Then

£\y,< := {Qi,^\3Vi^e£ : Qn = {Yn}Wn} is an admissible class for (Y1œ, g1<<) and

ch£W< (g1<) < h£(fi,<).

If c = 1, then equality holds.

Proof. It is clear that £\Y1( satisfies Axiom (A). Let Q1( E £1((\Y1oo. Then there exists P1( E £ such that the elements of each Qn are the intersections of the elements of Pn with Yn. Now assume that is a sequence of

partitions for Y1( which is coarser than Q1(. Then the elements of each Rn are unions of elements of Qn. Taking corresponding unions of elements of Pn for each n, one constructs a sequence 51( E £ coarser than P1( such that {Yn} VS1(( = R1(, which proves that £\Y1 ( satisfies Axiom (B). Finally, if Qn = {Yn} VPn for some P1( E £, then for all k,m > 1 it holds that

m—1 m—1 m—1

V = V f—i({Y+k}VPi+k) = Y}V V f—P+k,

1=0 1=0 1=0

which implies that £\Y^( satisfies Axiom (C). To prove the inequality of entropies, consider Q1( E £\Y^( and the corresponding P1( E £ with Qn = {Yn} VPn. Then

Hvi V = Hi \{Y:}V V f-"P'

■- £ ¡i(P n Yi)log

¡i(P n Yi)

Pe\li f-"Pi+i

£ ¡i(P n Yi) log№(P n Yi) - Y- Vi(P n Yi) log C

.PeViff'Pi+i PeVif-'Pi+i

The last summand gives

¡J1(P n Y1)log c = ^1(Y1)log c = c log c,

PeVi f-'P'+i

and thus can be omitted in the computation of h(g1(1; Q1(). We obtain

h(gi,*>; Qi,™) = Um sup -

n —»TO n

£ ¡i(P n Yi) log¡i(P n Yi)

PeVi f-'P'+i

If we consider the sequence V1oO of partitions V„ := {P n Y„ : P G V„} U {P n Z„ : P G V„}, we see that

"\yi,00I =£1 ,OO J "\'1,OQ I Vi

By the assumption on £ it follows that V1oOX, G £ and hence the assertion follows. In the case c = 1, the measures fJ„(Z„J are all zero, and hence equality holds in (3). Since V1oOX, is finer than V1oOX,, we have

h£(fi,oJ = sup h(fi oI Vio) = c sup h(gioI Qi,oJ = ch£W {gi.oJ,

Vio Q1o G£\Yio

which finishes the proof. □

Remark 24.

For a topological NDS given by a sequence of homeomorphisms, endowed with an invariant sequence of Borel probability measures, the above proposition can be applied to the decomposition Y„ := supp¡j„, Z„ := X„\supp¡j„, where supp¡j„ = {x G X„\ie > 0 : )J„(B(x, eJJ > 0} is the support of the measure ¡j„.

3.4. The Power Rule for Metric Entropy

Given a metric NDS (X1oo, f1 OJ and k G N, we define the k-th power system (XfO, ffOoJ in exactly the same way as we did for topological systems. It is very easy to see that this system is a metric system as well.

If £ is an admissible class for (X1oo, f1ooJ, we denote by the class of all sequences of partitions for X1O which are defined by restricting the sequences in £ to the spaces in X^O, i.e., V1l00 = {V„} G £ iff

Vl^:= {V{„-i)k+i}„>i g£k

Proposition 25.

If £ is an admissible class for (X1oo, f1 OJ, then is an admissible class for (X^O, ffOoJ and

h£[k] (flt) = k • h£ (fi,o) . (4)

Proof. It is clear that satisfies Axiom (A). To verify Axiom (B), consider v\kO G £[k] for some V1o G £. If Q1o00 is a sequence of partitions for X^O which is coarser than vfO (i.e., Q„ < V„„—1Jk+1 for all „ > 1), we can extend Q1oo to a sequence of partitions for X1o which is coarser than V1o. This can be done in a trivial way by putting

^ V„ if „ — 1 is not a multiple of k,

Q1+(„-1]/k if „ — 1 is a multiple of k.

Then it follows that V„„ = V„ < V„ in the first case, and V„„ = Q1+(„—1]/k < V„ in the second one. Since £ satisfies Axiom (B), we know that K1o G £, which implies that Q1oo = fcfO G . To show that £k satisfies Axiom (C), let V1o G £ and m > 1. We have to show that the sequence Q1oO defined by

Q„ := V f{„%k+1V(i+„—1]k+1 i=0

is an element of £k. To this end, first note that

mk—1

Q„ ^ \/ f(„i—1)k+1V(„—1)k+1+i =: K„. i=0

The sequence can be extended to an element of £, which Is given by

sn := V f-iV' ¡=0

Indeed, G £, since £ satisfies Axiom (C). Hence, = 5] ^ G and since

satisfies Axiom (B), this implies

Qi,™ G £|k|. Now let us prove the formula for the entropies. Let Vi,<x G £. We define a sequence Q^ of finite measurable partitions for XH as follows:

Qn := V f--i)k+iV(n-i)k+i+j-j=o

The sequence Qi^ is an element of £[k], since it is of the form = R'H with R^ G £. This follows by combining the facts that Vi^ g£ and £ satisfies Axiom (C). We find that

h (C QiJ = lim SUP-H,i \\ f"'k Qi+1 J \ ' " \i=0 I

= lim sup " H,i ( \\ f"'k V f-J+iVkk+i+j ) n \i=0 i=0 j

= limsup " H„1 (\\ V fr"k+'Hk+„+i )

" \i=0 j=0 J

1 /"k-1 \ = k • lim sup —- Hm Y = k • h(fi^, Vi,J).

nk ^ ¡=0 I

To obtain the last equality we used Proposition 9 (lv). Now consider also the sequence VH. It Is obvious that Q1ê^ Is fin er than v1 ^. Hence, using Proposition 9 (iii), we find

h (C- v1kL) < h (f1kL; Qi.M) = k • h (fi_ Vi.„).

Taking the supremum over all vil on the left-hand side and over all Vi^ on the right-hand side, the inequality "<" in (4) follows. The converse inequality follows from

Vi (C) > h (f^; Qi.„) = k • h (*,„; Vi.„),

which holds for every Vi^ G £. □

4. Relation to Topological Entropy

In order to prove a variational inequality, we consider a topological NDS (Xi ^1,fi ^1) with an f| ^-invariant sequence fji <x of Borel probability measures. When speaking of measurable partitions in this context, we mean "exact" partitions and not partitions in the sense of measure theory, where different elements of the partition may have a nonempty overlap of measure zero. We will frequently use the property of inner regularity of Borel measures, i.e., ¡j(A) = sup{^(K) : K C A compact} for any Borel subset of a compact metric space.

4.1. The Misiurewicz Class

In this subsection, we Introduce a special admissible class which we will use to prove the variational Inequality. This class is constructed in such a way that its elements are just perfect to apply the arguments of Misiurewicz's proof of the variational principle to them. Therefore, we call it the Misiurewicz class.

Let (X1oo, f1sxi) be a topological NDS with an f1sxi-invariant sequence of Borel probability measures ¡j1sx, = {^„}. We define the Misiurewicz class £M C Emax as follows. A sequence V1i00 G Emax, T„ = {Pni, ■ ■ ■, Pn,k„}, is an element of em iff for every e > 0 there exist 5 > 0 and compact sets Cni C P„j (n > 1, 1 < i < kn) such that for every n > 1 the following two hypotheses are satisfied:

Proposition 26.

If f1iXI is equicontinuous, then EM is an admissible class.

Proof. First note that E^ is nonempty, since it contains the trivial sequence defined by Vn := {Xn} for all n > 1. To show that EM satisfies Axiom (B), assume that T1i00 = {Vn} G EM, Vn = {Pni, ■ ■ ■, Pn,kn}, and let Q1iXI be a sequence which is coarser than T1i00. Let Qn be given by

Since V1i00 G EM, we can choose compact sets Cn i C Pn,i and 5 > 0 depending on a given e = e/(maxn>1 #Vn) such that (a) and (b) hold for T1i00. Define

(a) Hn(Pn.\Cn.i) < e.

(b) The minimal distance between the sets Cn i is at least 5, i.e.,

min min {Q„(x, y) : (x, y) G Cn i x Cn j} > 5.

Qn = [Qua.....Qn,i„}.

Then every element of Qn must be a disjoint union of elements of Vn:

n > 1, i = 1.....I,

It is clear that Dn i is a compact subset of Qn i. Moreover, it holds that

maxna1 #V,

< £■

For i = j we have

= min min

im {Qn(x, y) : (x, y) G Cn,ja x Cn,jß} > 5,

since each Cnja is disjoint from all Cnj^. Hence, G £M. To show that Axiom (C) holds, let = {Vn} G £M, Vn = {Pni,..., Pn,k„}, and m > i. Consider the sequence V^(fi,cx>). For given e = (i /m)e > 0 choose 5 > 0 and compact sets Cn1 C Pn,i such that (a) and (b) hold for ViOT. For every r > i and (j0,..., jm-i) G {i,..., kr }x-x{i,..., kr+m-i } define

Dr,(j0.....jm-i) := fl f-'(Cr+Ui).

These sets are obviously compact subsets of Xr and each element of Vrm^(fi OT) contains exactly one such set. We have

Kr[ D f-'P^i )\ f| f"'(Cr+m )

¡=0 I

fl f-' P'+w H v^Cr+t.i, ) ¡=0

<£ Vr (frl(Pr+Ul )\f- (Cr+i.n )) i=0

m-1 m-1

= £ f^r ( Pr+l,jl\Cr+l,ji) =YL Vr+l ( Pr+l,jl\Cr+l,ji) < me = ?.

Finally, in order to show that (b) holds for Vm(fi,cx>), we need the assumption of equicontinuity for fi xi, which yields a number p > 0 such that Qr (x,y) < p implies Qr+i(fl:(x), fr'(y)) < 5 for all r > i and 1 = 0, i,...,m - i (cf. the

proof of Lemma 3). Now consider two sets Drj0.....jm-i) and Dr (i0.....¡m-i). These sets are disjoint iff there is an index

a G {0, i.....m - i } such that ja = la. This implies Qr+a№(x), fa(y)) > 5 for all x G D^.....¡m-i) and y G D^.....m-i),

and hence Qr(x, y) > p. Thus, we have found that for every r > i it holds that

ITlin min {Qr(x, y) : (x, y) G Dr,(k,...Jm-1) X Dr,(l0.....lm_1)} > p,

......Jm-1)=

(l0.....lm-1 )

which completes the proof. □

In [13, Thm. B] it is shown that an equiconjugacy preserves the topological entropy of a topological NDS. An equicon-jugacy between systems f1œ and g1ixi is an equicontinuous sequence = {nn} of homeomorphisms such that also {n-1} is equicontinuous and nn+1 o fn = gn o nn. The following proposition shows that an equiconjugacy also preserves the Misiurewicz class and hence the associated metric entropy.

Proposition 27.

Consider two equicontinuous topological NDSs (X1œ, f1œ) and (Y1œ, g1,œ). Assume that is an equisemiconjugacy from f1œ to g1œ, i.e., it holds that nn+1 o fn = gn onn for all n > 1 and the sequence {nn} is equicontinuous. Then, if is an f1 ^-invariant sequence, v1œ = {vn}, vn := nn^n, is g1 ^-invariant and is an £M(f1^1)-EM(g1^1)-semiconjugacy. Hence,

h£M(g1,™) < h«M(f1,~).

Proof. We have gnvn = gn(nn^n) = nn+1fnvn = nn+1 ¡jn+1 = vn+1 and hence, v1œ is g1œ-invariant. To show that is an fM(f1,™)-fM(g1,™)-semiconjugacy, consider some G £u(g1,^) and let Vn := {n-1(Ç) : Q G Qn} for all

n > 1. To show that V1,TO G £M(f1lXI), let e > 0. Then, if Qn = {Qn1,..., Qn,kn}, we find compact sets Cnj C Qn,' and 5 > 0 such that vn(Qnj\Cnj) < e and

^min^ min {qy" (y1,y2) : (y1,y2) G Cm x Cnj} > 5. (5)

Since {nn} Is equlcontlnuous, there exists p> 0 such that QXn (x1,x2) < P Implies qy„ (nn(x1), nn(x2)) < 5 for all n > 1 and x1,x2 e Xn. Now consider the closed (and hence compact) sets 7T-1(Cni) C n-1 (Qni) =: Pni e Vn. We have )Jn(Pn,i\n-1 (Cn,i)) = vn(Qnii\Cn i) < e. Assume to the contrary that there exist n e N, i = j, and x1 e JT-1(Cn i), X2 G TT-1(Cn,j), such that Qxn (xi,x2) < p. This implies QYn (nn(xi), nn(x2)) < 5. Since nn(xi) e Cn,i and nn(x-2) e Cnj this contradicts (5). Hence, V1i00 e £M(f1l0o) and the rest follows from Proposition 22. □

4.2. The Variational Inequality

Now we are in position to prove the general variational inequality following the lines of Misiurewicz's proof [19].

Theorem 28.

For an equicontinuous topological NDS (X1oo, f1oo) with an invariant sequence ¡j1oo it holds that

h£M(f1,<x) < htop(fi <x).

Proof. Let V1i00 e Em. We may assume that each Vn has the same number k of elements, Vn = {Pn,t,..., Pn,k}. By definition of the Misiurewicz class, we find compact sets Qn i C Pn,i (for all n, i) such that

MPn.i\Qn,i) <

i = 1.....k, n > 1,

and 5 > 0 with

1<i<j<k

k log k '

i^Qn{x, y) : (x, y) e Qni x Qn,j} > 5.

By setting Qn0 := Xn\Uk=1 Qni we can define another sequence Q1:00 of measurable partitions Qn : = {Qn,0, Qni,..., Qn,k}. As in Misiurewicz's proof one finds Hfln (Vn\Qn) < 1, which by Proposition 9 (vi) leads to the inequality

h (fi,»; Vi^) < h (fi^; Qi^) +1. (7)

Define a sequence U1sxi of open covers Un of Xn by

Un := {Qn,o U Qni.....Qn,o U Qn,k} .

To see that the sets Qn 0 U Qn i are open, consider their complements Qn1 U ... U Qn,i-1 U Qn,i+1 U ... U Qn k, which are finite unions of compact sets and hence closed. For a fixed m > 1, let Em C X1 be a maximal (m, 5)-separated set. From (6) it follows that each (5/2)-ball in Xn intersects at most two elements of Qn for any n > 1. Hence, we can associate to each x e Em at most 2m different elements of \/m=0 f1-'Qi+1, which implies

< 2mrsep ( m, 2,fi,o

Consequently, we obtain

V f-'Q^} < 1og# Using (7), we therefore have

< log ^sep ( m, 2,fi,o ) + m log2.

h (fio Vi,<x ) < lim sup —log ^p (m,5,fi:<x J + log 2 + 1

m^ oo m \ 2 /

< htop(fio) + log2 + 1.

Taking the supremum over all V1ê<x G ¿M, we find

hsM(fi,™) < htop(fi^) + log2 + 1.

That the constant term log 2 + 1 can be omitted in this estimate now follows from a careful application of the power rules for topological and metric entropy. Inspecting the definition of the Misiurewicz class, one sees that for every k > 1 the admissible class ¿M is contained in the Misiuri system (X1œ, f1iXI) can equally be applied to a (Proposition 2 and Proposition 25), we obtain

admissible class fM Is contained In the Misiurewicz class of f\k'jx. Therefore, the arguments that we have applied to the system (X1œ, f1œ) can equally be applied to all of the power systems (XH, fH), k > 1. Hence, using the power rules

u it \ ^ u it \ log 2 +1

ll£M(ti,c) < "top(M,c) +--^-•

Since this holds for every k > 1, sending k to Infinity gives the result. □

An interesting corollary of Theorem 28 is the following generalized variational principle for autonomous systems.

Corollary 29.

For a topological autonomous system (X, f) it holds that

suP h£M(f.m.o)(f) = htop(f),

where the supremum is taken over all sequences Vt,c with fvn = Vn+t.

Proof. The inequality "<" holds by Theorem 28. The converse inequality follows from the classical variational principle, if we consider only the constant sequences V1c, i.e., the invariant measures of f, and assure ourselves that the associated Misiurewicz classes contain all constant sequences. □

Corollary 30.

Let f1c be an equicontinuous sequence of (not necessarily strictly) monotone maps fn : X ^ X, where X is either a compact interval or a circle. Then for every f1ci-invariant sequence v1c it holds that hgM(f1ci) = 0.

Proof. This follows from [13, Thm. D], which asserts that the corresponding topological entropy is zero. □

4.3. Large Misiurewicz Classes

Up to now, we only know that the Misiurewicz class contains the trivial sequence of partitions. If it would contain no further sequences, Theorem 28 would not give any valuable information on the metric or topological entropy. The aim of this subsection is to find conditions on invariant sequences of measures which give rise to a large Misiurewicz class. The simplest case consists in a system (X1cC,f1cC,v1c), where both X1c and v1c are constant, say Xn = X and Vn = V. Then any finite measurable partition V of X gives rise to a constant sequence Vn = V of partitions which is obviously contained in £M. The following proposition slightly generalizes this situation.

Proposition 31.

Let (X1c, f1c) be an equicontinuous NDS with an f1cC-invariant sequence Vt,c. If X1c is constant and the closure of {Vn} with respect to the strong topology on the space of probability measures is compact, then contains all constant sequences of partitions.

Proof. We first show that every Borel set A C X can be approximated by compact subsets uniformly for all jn. The strong topology is characterized by

¡jn — j ^ ¡jn(A) — /j(A) for every Borel set A C X.

Let C be the strong closure of j1oo, and let A C X be a Borel set and e > 0. For each j eC there exists a compact set BJ C A such that j(A\BJ) < e/2. Now take a neighborhood Uj of j in C such that \v(A\Bj) - j(A\Bj)\ < e/2 for all v e Uj. Then for every v eUJ we have

v(A\Bj) < j(A\Bj) + 2 < e.

We can cover the compact set C by finitely many of such neighborhoods, say Uj1,..., Ujr. Then B : = |J [=1 Bji is a compact subset of A which satisfies v(A\B) < e for all v e C, so in particular for all v = jn. Now let V = {P1,..., Pk} be a finite measurable partition of the state space X. Then for any given e > 0 we find compact sets C' C P' such that jn(P'\C') < e for all n > 1 and i = 1,..., k. Moreover, since the sets C' are pairwisely disjoint,

min min {o(x, u) : (x, u) e C' x C^ > 0.

1<'<j<k "

This implies that the constant sequence Vn =V is an element of £M. □

Example 32.

Consider a system which is given by a periodic sequence

f1,co = {f1,f2,...,fN,f1,f2,...,fN,...} .

Let j1 be an fN-invariant probability measure on X (which exists by the theorem of Krylov-Bogolyubov). Define

J2:= f1/1, J3 := f2/2..... JN := fN-1/N-1,

and extend this to an N-periodic sequence

j1,oo = {J1,J2.....JN,J1 ,J2.....JN,...} .

Then j1oo is an f1oo-invariant sequence, which follows from

fNjN = fNfN-1jN-1 = fNfN-1fN-2jN-2 = • • • = fNj1 = J1 . Clearly, {j1,..., jN} is compact.

The assumption that the closure of {jn} should be compact still seems to be very restrictive. The next result (Proposition 34) provides another condition for a large Misiurewicz class.

Lemma 33.

Let (X, q) be a compact metric space with a Borel probability measure j. Let A C X be a Borel set with j(dA) = 0. Then A can be approximated by compact subsets with zero boundaries, i.e.,

j(A) = sup {j(K) : K C A compact with j(dK) = 0} .

Proof. We can assume without loss of generality that dA = 0, since otherwise A is closed and hence compact itself. For every e > 0 define the set

Ke := {x e int A : dist(x, dA) > e} •

We claim that each Ke is a closed subset of X and hence compact. To this end, consider a sequence xn e Ke with xn — x e X. By continuity of dist( ,dA), it follows that dist(x, dA) > e and x e clA. Assume to the contrary that x e dA. Then e < dist(x, dA) = 0, a contradiction. Hence, x e Ke. We further claim that v(Ke) —> v(A) for e — 0. To show this, take an arbitrary strictly decreasing sequence en —> 0. Then Ken C Ken+1 for all n > 1. Hence, by continuity of the measure v and the assumption that v(dA) = 0, it follows that

V(A) = v(int A) = v | J KeJ = Um v(Ke„)•

To conclude the proof, it suffices to show that one can choose the sequence en such that v(dKen) = 0. To this end, we first show that for 51 < 52 the boundaries of K^ and K^2 are disjoint. Assume to the contrary that there exists x e dKfr1 n dKi2. Then, by continuity of the dist-function, dist(x, dA) > 51 and dist(x,dA) > 52. However, if one of these inequalities would be strict, the point x would be contained in the interior of the corresponding set. Hence, dist(x,dA) = 51 < 52 = dist(x,dA), a contradiction. Now, we can construct the desired sequence en — 0 as follows. Fix n e N and assume to the contrary that v(dKe) > 0 for all e e (1 /(n +1), 1 /n). Define the sets Im := {e e (1 /(n +1), 1 /n) : v(dKe) > 1/m}. Then (1/(n + 1), 1/n) = IJmeN Im and hence one of the sets Im, say Im0, must be uncountable. However, since the boundaries of the Ke are disjoint, this would imply that the set IJdKe has an infinite measure. Hence, we can take en e (1/(n +1), 1/n) with v(dKen) = 0. □

Proposition 34.

Let (X1c, f1c) be an equicontinuous system such that X1c is constant and let Vt,c = {vn} be an f1 ^-invariant sequence. Assume that the measures in the weak*-closure of {vn} are pairwisely equivalent. Then £M contains all constant sequences of partitions whose members have zero boundaries (with respect to the measures Vn).

Proof. Let C denote the weak*-closure of {vn}. Consider a finite measurable partition V = {P1, • • •, Pk} of the state space X such that v(dp) = 0, 1 < i < k, for one and hence all v eC. Fix e > 0 and pick v e C. By Lemma 33, we find compact sets Cvi C Pi with v(dCvi) = 0, 1 < i < k, and

v(Pi\CVi) < e/2, 1 < i < k.

Since d(Pi\Cv i) C 3Pi U dCv i and hence v(d(Pi\Cv i)) = 0, the Portmanteau theorem yields a weak*-neighborhood Uv C C of v such that for every v eUv it holds that \ v(Pi\CvA) - v(P\Cv,i)\ < e/2. Therefore, v(P\Cv,i) < e for all V e Uv. Since C is weakly*-compact, we can cover C with finitely many of these neighborhoods, say UV1, • • •,UVr. Then C[ := U-=1 CVi is a compact subset of Pi for 1 < i < k and for every v e C it holds that v(P\Ci) < e, in particular for all v = vn. This implies that the constant sequence Vn =V is in £M. □

Remark 35.

Note that every compact metric space admits finite measurable partitions of sets with arbitrarily small diameters and zero boundaries (cf. [11, Lem. 4.5.1]).

Example 36.

An example for systems with invariant sequences satisfying the assumption of Proposition 34, can be found in [23]: Let M be a compact connected Riemannian manifold. By d(•, •) denote the Riemannian distance and by m the Riemannian volume measure. For simplicity, we will assume that m(M) = 1, so m is a probability measure. For constants A> 1 and r > 0 consider the set

£ (A, r) := {f eC2(M,M) : f expanding with factor A, \\f\\C2 < T} ,

where "expanding with factor A" means that \Dfx(v)| > A\v\ holds for all x e M and all tangent vectors v e TxM. We will consider a NDS = {fn} on M with fn e £ (A, r) for fixed A > 1 and r > 0. It Is clear that such a system Is equlcontlnuous. We define

and for every L > 0 the set

p : M R : p> 0, Llpschltz, J pdm = 1

p eV :

p(x) p(y)

< Ld(x, y) if d(x, y) < e

where e > 0 is a fixed number (depending on A and r). Note that

V = U n,

since for every p eV we have

p-)\pix ) - p(y)\ < mnp d(x,y).

For any expanding map f : M ^ M we write

V, (p)(x) = ^

- \ det Df(y)\,

yef 1(x)

Vf(p) : M ^ R,

for the Perron-Frobenius operator associated with f acting on densities p e V. Note that this makes sense, since expanding maps are covering maps, and hence the sets f~1(x) are finite, all having the same number of elements. Now let p eV. We claim that the f1 ,^-invariant sequence, defined by ¡j1 := pdm and ¡jn := for all n > 2, has

the property that the elements of the weak*-closure of {^n}neN are pairwisely equivalent. To show this, let L > 0 be chosen such that p e VL and note that ¡jn+1 = Vfn (p)dm for all n. By [23, Prop. 2.3], there exist L* > 0 and t > 1 such that Vfn (p) e V** for all n > t. Hence, we may assume that Vfn (p) e V* for all n. We will first show that the densities in TL* are uniformly bounded away from zero and infinity and that they are equicontinuous. Assume to the contrary that there are pn e VL* and xn e M such that pn(xn) > n. Without loss of generality, we may assume that pn(xn) = maxxeM pn(x). Choosing 5 e (0, e] with L5 < 1, we obtain

/ p„dm > Jm Jb

1 = I pndm > J pn(x)dm(x) =

B(x„,5)

pn (x)

B(x„,5) pn(xn)

pn(xn)dm(x )

> ni (1 - Ld(x,xn))dm(x)

JB(x„,ö)

> ni (1 - L5) dm = n (1 - L5) m(B(xn,5 )).

JBIxn.i)

Since m(B(xn, 5)) is bounded away from zero, this is a contradiction. Hence, the functions in TL* are uniformly bounded by some constant K. This immediately implies equicontinuity, since for x, y G M with d(x, y) < e we have

\p(x) - p(y)\ = p(y)

p(x ) p(y)

< KLd(x, y).

To show that the p e VL* are uniformly bounded away from zero, assume to the contrary that there exist pn e VL* and xn e M such that pn(xn) ^ 0. By compactness, we may assume that xn ^ x. Then

\pn(x) - pn(xn)\ < KLd(x,xn) 0

Now pick some y G B(x, e). Then

\pn(x) - Pn(y)\ = pn(x)

< pn(x)Le — 0.

Using the theorem of Arzela-Ascoli, we can choose a uniformly convergent subsequence pmn —> p. The above argument shows that the closed set p-1(0) is open, and by assumption it is nonempty. Hence, it is equal to M. This is a contradiction to the integral condition pmndm = 1, which implies pdm = 1. Now we prove the claim: Let v be a weak* limit point of {vn} and let p1 := p, pn+1 := Vf^p. Then, for a subsequence mn and for every continuous g : M — R we have

/ gPmndm JM

On the other hand, by the theorem of /Arzela-Ascoli, we may assume that pmn converges uniformly to some p*, which is bounded away from zero and satisfies J p*dm = 1. Hence,

I gPm„dm — I gp*dm, Jm Jm

implying v = p*dm.

Remark 37.

The above example can be regarded as a nonautonomous version of the classical result of Krzyzewski and Szlenk [15] which asserts that every expanding C2-map has an absolutely continuous invariant measure.

Remark 38.

In view of Proposition 34 and Proposition 27, the most general criterion which guarantees a large Misiurewicz class for an equicontinuous system (X1^1,f1i^1) with invariant sequence ¡j1<x is the existence of an equiconjugacy to a system which satisfies the assumptions of Proposition 34. That is, there exists a compact metric space X and an equicontinuous sequence {nn} of homeomorphisms nn : Xn —> X such that all elements of the weak*-closure of the set {nn^n} are equivalent.

5. Concluding Remarks and Open Questions

In this paper, we introduced a notion of metric entropy for quite general nonautonomous dynamical systems and studied its elementary properties, in particular its relation to the topological entropy defined by Kolyada, Misiurewicz, and Snoha. The number of open questions about this new quantity tends to infinity. We restrict ourselves to a very short list of questions and topics for future research:

• In order to obtain a fruitful theory of metric entropy for nonautonomous systems, it seems inevitable to find appropriate analogues of the notion of ergodicity. Describing ergodicity as the property that the state space cannot be broken apart into two invariant subsets of positive measure, one can use the same definition for a metric NDS on a single probability space. However, this definition is probably too strict. It seems more likely that for different purposes different analogues of ergodicity of varying strength will fit.

• One of the next steps in the further development of the entropy theory for nonautonomous systems certainly is the study of the question to which extent the variational inequality (Theorem 28) can be extended to a full variational principle. Another interesting question is under which conditions there exist reasonably small generating sets for the Misiurewicz class.

• The classical Pesin formula and Margulis-Ruelle inequality relate the metric entropy of a diffeomorphism to its Lyapunov exponents, given by the Multiplicative Ergodic Theorem. It is an interesting and probably very far-reaching question to which extent such results can be transferred to the nonautonomous case.

• The notion of metric entropy in this paper also generalizes the metric sequence entropy introduced in Kushnirenko [16]. It might be an interesting topic for future research to look for generalizations of the known results about metric sequence entropy.

Acknowledgements

My gratitude is to Tomasz Downarowicz who pointed out Example 18 to me, and to the anonymous referee who made several good suggestions which helped me to improve the paper and who also brought some interesting literature to my attention. Furthermore, I acknowledge the financial support by DFG grant Co 124/17-2 within DFG priority program 1305.

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