Petr Kůrka

Limit Sets of Cellular Automata Associated to Probability Measures

We introduce the concept of limit set associated to a cellular automaton (CA) and a shift invariant probability measure. This is a subshift whose forbidden blocks are exactly those, whose probabilities tend to zero as time tends to infinity. We compare this probabilistic concept of limit set with the concepts of attractors, both in topological and measure-theoretic sense. We also compare this notion with that of topological limit set in different dynamical situations.

Subshift attractors of cellular automata

A subshift attractor is a two-sided subshift which is an attractor of a cellular automaton. We prove that each subshift attractor is chain-mixing, contains a configuration which is both F-periodic and σ-periodic and the complement of its language is recursively enumerable. We prove that a subshift of finite type is an attractor of a cellular automaton iff it is mixing. We identify a class of chain-mixing sofic subshifts which are not subshift attractors. We construct a cellular automaton whose maximal attractor is a non-sofic mixing subshift, answering a question raised in Maass (1995 Ergod. Theory Dyn. Syst. 15 663–84). We show that a cellular automaton is surjective on its small quasi-attractor which is the non-empty intersection of all subshift attractors of all Fqσp, where q > 0 and .

Möbius number systems with sofic subshifts

A real M?bius iterative system is an action of a free semigroup of finite words acting via M?bius transformations on the extended real line. Its convergence space consists of all infinite words, such that the images of the Cauchy measure by the prefixes of the word converge to a point measure. A M?bius number system consists of a M?bius iterative system and a subshift included in the convergence space, such that any point measure can be obtained as the limit of some word of the subshift. We give some sufficient conditions on sofic subshifts to form M?bius number systems. We apply our theory to several number systems based on continued fractions.

Analytical evidence for scale-invariance in the shape of species abundance distributions

The distribution of species abundances within an ecological community provides a window into ecological processes and has important applications in conservation biology as an indicator of disturbance. Previous work indicates that species abundance distributions might be independent of the scales at which they are measured which has implications for data interpretation. Here we formulate an analytically tractable model for the species abundance distribution at different scales and discuss the biological relevance of its assumptions. Our model shows that as scale increases, the shape of the species abundance distribution converges to a particular shape given uniquely by the Jaccard index of spatial species turnover and by a parameter for the spatial correlation of abundances. Our model indicates that the shape of the species abundance distribution is taxon specific but does not depend on sample area, provided this area is large. We conclude that the species abundance distribution may indeed serve as an indicator of disturbances affecting species spatial turnover and that the assumption of conservation of energy in ecosystems, which is part of the Maximum Entropy approach, should be re-evaluated.

The Stern–Brocot graph in Möbius number systems

We characterize interval Mobius number systems with sofic expansion subshifts and show that they can be obtained as factors of interval Mobius number systems with expansion subshifts of finite types. The endpoints of interval cylinders of such systems can be computed by a simple formula which generalizes the computation of Farey fractions in the Stern–Brocot graph. We treat in detail the bimodular number system which has many nice properties and could be used for exact real computer arithmetic.

Möbius number systems based on interval covers

Given a finite alphabet A, a system of real orientation-preserving Mobius transformations , a subshift and an interval cover of , we consider the expansion subshift of all expansions of real numbers with respect to . If the expansion quotient is greater than 1 then there exists a continuous and surjective symbolic mapping and we say that is a Mobius number system. We apply our theory to the system of binary continued fractions which is a combination of the binary signed system with the continued fractions, and to the binary square system whose transformations have stable fixed points −1, 0, 1 and ∞.

A symbolic representation of the real Möbius group

We describe symbolic representations of the extended real line based on the dynamical systems consisting of Mobius transformations. The representations can be extended to the group of real Mobius transformations.

Language complexity of rotations and Sturmian sequences

Given a rotation of the circle, we study the complexity of formal languages that are generated by the itineraries of interval covers. These languages are regular iff the rotation is rational. In the case of irrational rotations, our study reduces to that of the language complexity of the corresponding Sturmian sequences. We show that for a large class of irrationals, including e, all quadratic numbers and more generally all Hurwitz numbers, the corresponding languages can be recognized by a nondeterministic Turing machine in linear time (in other words, belongs to NLIN).

Finite state transducers for modular möbius number systems

Modular Mobius number systems consist of Mobius transformations with integer coefficients and unit determinant. We show that in any modular Mobius number system, the computation of a Mobius transformation with integer coefficients can be performed by a finite state transducer and has linear time complexity. As a byproduct we show that every modular Mobius number system has the expansion subshift of finite type.

Game dynamics and evolutionary transitions

An evolutionary model based on the Taylor-Jonker game dynamics is presented. A set of strategies is compatible if there exists a dynamical equilibrium between its members and there is an evolutionary transition to another compatible set if new mutant strategies bring about a passage to another equilibrium. We apply these concepts to supergame strategies, which play repeatedly a given matrix game and at each time step choose their pure strategy according to the preceding moves of the opponent. We investigate the patterns of evolution in zero-sum games, games of partnership, the prisoners dilemma and the hawkdove game.