A Combinatorial classification of postcritically fixed Newton maps

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to $\infty$ through a finite chain of such components.Comment: 37 pages, 5 figures, published in Ergodic Theory and Dynamical Systems (2018). This is the final author file before publication. Text overlap with earlier arxiv file observed by arxiv system relates to an earlier version that was erroneously uploaded separately. arXiv admin note: text overlap with arXiv:math/070117

Cost-Bounded Active Classification Using Partially Observable Markov Decision Processes

Active classification, i.e., the sequential decision-making process aimed at data acquisition for classification purposes, arises naturally in many applications, including medical diagnosis, intrusion detection, and object tracking. In this work, we study the problem of actively classifying dynamical systems with a finite set of Markov decision process (MDP) models. We are interested in finding strategies that actively interact with the dynamical system, and observe its reactions so that the true model is determined efficiently with high confidence. To this end, we present a decision-theoretic framework based on partially observable Markov decision processes (POMDPs). The proposed framework relies on assigning a classification belief (a probability distribution) to each candidate MDP model. Given an initial belief, some misclassification probabilities, a cost bound, and a finite time horizon, we design POMDP strategies leading to classification decisions. We present two different approaches to find such strategies. The first approach computes the optimal strategy "exactly" using value iteration. To overcome the computational complexity of finding exact solutions, the second approach is based on adaptive sampling to approximate the optimal probability of reaching a classification decision. We illustrate the proposed methodology using two examples from medical diagnosis and intruder detection

Complete controllability of finite-level quantum systems

Complete controllability is a fundamental issue in the field of control of quantum systems, not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables. In this paper we investigate the question of complete controllability for finite-level quantum systems subject to a single control field, for which the interaction is of dipole form. Sufficient criteria for complete controllability of a wide range of finite-level quantum systems are established and the question of limits of complete controllability is addressed. Finally, the results are applied to give a classification of complete controllability for four-level systems.Comment: 14 pages, IoP-LaTe

Almost isomorphism for countable state Markov shifts

Countable state Markov shifts are a natural generalization of the well-known subshifts of finite type. They are the subject of current research both for their own sake and as models for smooth dynamical systems. In this paper, we investigate their almost isomorphism and entropy conjugacy and obtain a complete classification for the especially important class of strongly positive recurrent Markov shifts. This gives a complete classification up to entropy conjugacy of the natural extensions of smooth entropy expanding maps, including all smooth interval maps with non-zero topological entropy

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24 versio

Morphisms, Symbolic sequences, and their Standard Forms

Morphisms are homomorphisms under the concatenation operation of the set of words over a finite set. Changing the elements of the finite set does not essentially change the morphism. We propose a way to select a unique representing member out of all these morphisms. This has applications to the classification of the shift dynamical systems generated by morphisms. In a similar way, we propose the selection of a representing sequence out of the class of symbolic sequences over an alphabet of fixed cardinality. Both methods are useful for the storing of symbolic sequences in databases, like The On-Line Encyclopedia of Integer Sequences. We illustrate our proposals with the $k$-symbol Fibonacci sequences