Abstract. We study dynamical properties of a parameterized family of quadratic rational maps, all of whose members lack period 2 orbits. We classify regions in the parameter space of the family according to the behavior of marked critical points. We characterize the parameter space by comparing it with the Mandelbrot set.

Topological feedback entropy (TFE) measures the intrinsic rate at which a continuous, fully observed, deterministic control system generates information for controlled set-invariance. In this paper we generalise this notion in two directions; one is to continuous, partially observed systems, and the other is to discontinuous, fully observed systems. In each case we show that the corresponding generalised TFE coincides with the smallest feedback bit rate that allows a form of controlled invariance to be achieved.Keywords Topological entropy · communication-limited control · quantised systems 1 Introduction In 1965, Adler, Konheim, and McAndrew [1] introduced topological entropy as a measure of the fastest rate at which a continuous, discrete-time, dynamical system in a compact space generates initial-state information. Though related to the measuretheoretic notion of Kolmogorov-Sinai entropy (see e.g. [13]), it is a purely deterministic notion and requires only a topology on the state space, not an invariant measure. Subsequently, Bowen [3] and Dinaburg [6] proposed an alternative, metric based definition of topological entropy. This accommodates uniformly continuous dynamics on noncompact spaces and is equivalent to the original definition on compact spaces.These concepts play an important role in dynamical systems but remained largely neglected in control theory. However, the emergence of digitally networked control systems (see e.g. [2]) over the last four decades renewed interest in the information theory of feedback, and in 2004 the techniques of Adler et al. were adapted to introduce the notion of topological feedback entropy (TFE) [11]. Unlike topological

The critical dimension is an invariant that measures the growth rate of the sums of Radon–Nikodym derivatives for non-singular dynamical systems. We show that for Bratteli–Vershik systems with multiple edges, the critical dimension can be computed by a formula analogous to the Shannon–McMillan–Breiman theorem. This extends earlier results of Dooley and Mortiss on computing the critical dimensions for product and Markov odometers on infinite product spaces.

Abstract. We study a family of rational maps of the sphere with the property that each map has two fixed points with multiplier −1; moreover each map has no period 2 orbits. The family we analyze is ( ) =, where varies over all nonzero complex numbers. We discuss many dynamical properties of including bifurcations of critical orbit behavior as varies, connectivity of the Julia set ( ), and we give estimates on the Hausdorff dimension of ( ).

In this paper, we construct an example of a cylindrical measure µ and a norm · such that · is µ-measurable by Dudley et al. and not µ-measurable by Gross and moreover is γ -measurable.