Metric Invariance Entropy and Conditionally Invariant Measures

A notion of metric invariance entropy is constructed with respect to a conditionally invariant measure for control systems in discrete time. It is shown that the metric invariance entropy is invariant under conjugacies, the power rule holds, and the (topological) invariance entropy provides an upper bound. November 20, 2014 Key words. invariance entropy, conditionally invariant measures, quasi-stationary measures, control systems MSC 2010: 37A35, 37B40, 93C10, 94A17 1. Introduction. This paper introduces and studies a notion of metric invariance entropy in analogy to the topological notion of invariance entropy of deterministic control systems, cf. Nair, Evans, Mareels, and Moran [12], and the monograph Kawan [9]. We consider control systems in discrete time of the form xn+1 = f(xn; un); n 2 N0 := f0; 1; : : :g; (1.1) where f : M ! M is continuous and M and are metric spaces. Abbreviate f! := f( ; !) : M !M for ! 2 , and for u = (un)n2Z 2 U := Z write the solutions as ' : N0 M U !M; '(0; x; u) := x; '(n; x; u) := fun 1 fu0(x) for n 1: The system should be kept in a compact subset Q of M . (In the literature there are colorful terms to describe this situation: One may think of M n Q as a “trap” or as a “hole” in the state space or leaving Q means “killing” the system). The topological notion of invariance entropy hinv(Q) of a compact subset Q M describes the average data rate needed to keep the system in Q. It is constructed with some analogy to topological entropy of dynamical systems. This is done in Nair, Evans, Mareels, and Moran [12] via the version of Adler, Konheim and McAndrews [1], and in Kawan [9, 8] via the version due to Bowen and Dinaburg based on spanning sets; cf., e.g., Downarowicz [5] for the entropy theory of dynamical systems. In the beginning of Section 3 a precise de…nition of invariance entropy in the setting of [12], called topological feedback entropy, will be given (actually, we use the slightly streamlined version from Colonius, Kawan, Nair [3]). The main di¤erence of entropy in a control context to entropy notions for dynamical systems is that the minimal required entropy is of interest, instead of the entropy generated by the system. If one wants to construct a metric entropy, the choice of an appropriate probability measure is crucial. The present paper proposes to use conditionally invariant measures for this purpose. In the dynamical systems literature, conditionally invariant (also called relatively invariant) measures have been introduced by Pianigiani and Yorke [13]; cf. the survey Demers and Young [4] and also Keller and Liverani [10]. For random systems, the related notion of quasi-stationary distributions (or measures) are Supported by DFG grant Co 124/19-1 and Brazilian European Partnership in Dynamical Systems FP7-PEOPLE-2012-IRSES 318999 BREUDS (EU) 1 a classical subject; cf. the recent monograph Collett, Martinez, and San Martin [2] and also Zmarrou and Homburg [15]. Intuitively, quasi-stationary measures describe “the distribution of trajectories which are on the verge of falling in the trap" [2, p. 15]. Control system (1.1) may be viewed as a skew product dynamical system by considering the left shift on U given by ( u)n := un+1; n 2 Z for u = (un) 2 U : Then S : (u; x) 7! ( u; f(x; u0)) is a skew product map on U M and its iterations de…ne a skew product dynamical system. Note that the product topology makes U = Z into a compact metrizable space. If one wants to keep system (1.1) in a compact subset Q M it appears appropriate to look at conditionally invariant measures for S on U M with respect to U Q. We construct a metric invariance entropy with respect to such a conditionally invariant measure. It will be shown that the metric invariance entropy is invariant under appropriately de…ned conjugacies, the power rule holds, and the topological invariance entropy is an upper bound. The main contribution of the present paper is the construction of a metric invariance entropy. The existence of quasi-stationary measures is brie‡y discussed in Section 2; they yield special conditionally invariant measures. The constructions for the metric invariance entropy given below in Sections 3 and 4 are conveniently done for general conditionally invariant measures. In the monograph by Collett, Martinez, and San Martin other su¢ cient conditions for the existence of quasi-stationary measures are derived; cf. [2, Proposition 2.10 and Theorem 2.11]. Demers and Young [4] discuss conditionally invariant measures, mainly for deterministic maps and with regard to absolutely continuous conditionally invariant measures and to escape rates. The contents of this paper is as follows: Section 2 discusses conditionally invariant measures for maps and for control systems; here also quasi-stationary measures are considered and notation is …xed. Section 3 constructs the metric invariance entropy and proves basic properties. Finally, Section 4 shows that the topological notion of invariance entropy provides an upper bound for the metric invariance entropies. 2. Conditionally invariant measures. In this section we collect some basic information on conditionally invariant measures and …x some notation. For a map S : X ! X on a metric space with metric d and A X we let S A := fx 2 X j S(x) 2 Ag. Definition 2.1. Let S : X ! X be a continuous map on a metric space X and consider a compact subset Y X. A probability measure on X endowed with the Borel -algebra B(X) is called conditionally invariant with respect to Y with constant if 0 < := (S Y \ Y ) 1 and (A) = (S A \ Y ) (S 1Y \ Y ) for all A 2 B(X). Putting A = Y in De…nition 2.1 one sees that the support of given by supp := fx 2 X j (N) > 0 for each open set with x 2 Ng is contained in Y . Hence, if we identify the probability measures on B(X) which have support in Y with the set P(Y ) of probability measures on the Borel -algebra B(Y ), a measure 2 P(Y ) is conditionally invariant if and only if (S Y ) > 0 and (A) = (S A) (S 1Y ) for all A 2 B(Y ).