Automatic generation of nonperiodic patterns from dynamical systems

A new and fast algorithm is presented in this paper for the automatic generation of aesthetic patterns on nonperiodic tilings by means of dynamical systems. The Chair and the Sphinx tilings are used as illustrations. Invariant mappings are constructed for the creation of striking patterns on these tilings. A modified convergence time scheme is described to enhance the artistic appeal of generated images. This algorithm can be used to create a great variety of exotic patterns with various nonperiodicities.     

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

An Algebraic Invariant for Substitution Tiling Systems

We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.     

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

   Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.     

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.     

Space tilings and substitutions

We generalize the study of symbolic dynamical systems of finite type and ² action, and the associated use of symbolic substitution dynamical systems, to dynamical systems with ² action. The new systems are associated with tilings of the plane. We generalize the classical technique of the matrix of a substitution to include the geometrical information needed to study tilings, and we utilize rotation invariance to eliminate discrete spectrum. As an example we prove that the pinwheel tilings have no discrete spectrum.Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grant 003658-113     

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.     

Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems

The concepts of collective sensitivity and compact-type collective sensitivity are introduced as stronger conditions than the traditional sensitivity for dynamical systems and Hausdorff locally compact second countable (HLCSC) dynamical systems, respectively. It is proved that sensitivity of the induced hyperspace system defined on the space of non-empty compact subsets or non-empty finite subsets (Vietoris topology) is equivalent to the collective sensitivity of the original system; sensitivity of the induced hyperspace system defined on the space of all non-empty closed subsets (hit-or-miss topology) is equivalent to the compact-type collective sensitivity of the original HLCSC system. Moreover, relations between these two concepts and other dynamics concepts that describe chaos are investigated.     

Continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system. Since the dynamical system is defined on an arbitrary Banach space (possibly infinite dimensional), to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by Lasota and Yorke and extended recently to infinite-dimensional spaces by Szarek. Finally, it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study. Hence we obtain their stability as well.     

All-time Morse decompositions of linear nonautonomous dynamical systems

Morse decompositions provide inside information about the global asymptotic behavior of dynamical systems on compact metric spaces. Recently, the existence of Morse decompositions for nonautonomous dynamical systems was proved by restricting attention to the past or the future of the system, but in general, such a construction is not realizable for the entire time. In this article, it is shown that all-time Morse decompositions can be defined for linear systems on the projective space. Moreover, the dynamical properties are discussed and an analogue to the Theorem of Selgrade is proved.

     

On the expressiveness and decidability of o-minimal hybrid systems

This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of o-minimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on o-minimal structures. In particular, we recover the main result of a paper by G. Lafferriere, G.J. Pappas, and S. Sastry, on o-minimal hybrid systems. We also provide an analysis and extension of results on decidability and complexity of problems and constructions related to o-minimal hybrid systems.     

Entropy increase in dynamical systems

The rate of increase of the non-equilibrium entropy introduced by Goldstein and Penrose, defined on nonstationary probability measures for an abstract dynamical system, is quantitatively related to the Kolmogorov-Sinai entropy of the system. It is shown in particular that for ergodic systems the asymptotic rate of entropy increase coincides with the Kolmogorov-Sinai entropy. Supported in part by NSF Grant No. PHY 78-03816.     

A spectral theory for linear differential systems

This paper is concerned with continuous and discrete linear skew-product dynamical systems including those generated by linear time-varying ordinary differential equations. The concept of spectrum is introduced for a linear skew-product dynamical system. In the case of a system of ordinary differential equations with constant coefficients the spectrum reduces to the real parts of the eigenvalues. In the general case continuous spectrum can occur and under certain conditions it consists of finitely many compact intervals of the real line, their number not exceeding the dimension of the system. A spectral decomposition theorem is proved which says that a certain naturally defined vector bundle is the sum of invariant subbundles, each one associated with a spectral subinterval. This partially generalizes the Jordan decomposition in the case of constant coefficients. A perturbation theorem is proved which says that nearby systems have spectra which are close. Almost periodic systems are given special attention.     

On local metric pressure of dynamical systems

In this paper, a local version of the topological pressure of dynamical systems is presented. It is a function defined on the product space which does not depend on any measure. It is shown that, for any invariant measure, integration of the introduced function with respect to its corresponding diagonal measure results in the metric pressure of the dynamical system.

     

A philosophy for the modelling of realistic nonlinear systems

A nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of realistic dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.

     

Tubes in dynamical systems

Important concepts concerning tubes in dynamical systems are defined in details. In the cases of a homoclinic tube and a heteroclinically tubular cycle in autonomous systems, existence of tubular chaos is established. The main goal of this article is to stress the importance of tubes in high dimensional dynamical systems.     

Change-of-bases abstractions for non-linear hybrid systems

We present abstraction techniques that transform a given non-linear dynamical system into a linear system, or more generally, an algebraic system described by polynomials of bounded degree, so that invariant properties of the resulting abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a change-of-bases transformation that associates each state variable of the abstract system with a function involving the state variables of the original system. We present conditions under which a given change-of-bases transformation for a non-linear system can define an abstraction. Furthermore, the techniques developed here apply to continuous systems defined by Ordinary Differential Equations (ODEs), discrete systems defined by transition systems and hybrid systems that combine continuous as well as discrete subsystems.The techniques presented here allow us to discover, given a non-linear system, if a change-of-bases transformation involving degree-bounded polynomials yielding an algebraic abstraction exists. If so, our technique yields the resulting abstract system, as well. Our techniques enable the use of analysis techniques for linear systems to infer invariants for non-linear systems. We present preliminary evidence of the practical feasibility of our ideas using a prototype implementation.     

A note on the structure of spaces of domino tilings

We study spaces of tilings, formed by tilings which are on a geodesic between two fixed tilings of the same domain (the distance is defined using local flips). We prove that each space of tilings is homeomorphic to an interval of tilings of a domain when flips are classically directed by height functions.     

Nilsequences and a structure theorem for topological dynamical systems

We characterize inverse limits of nilsystems in topological dynamics, via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems. We provide two applications of the structure. The first is to nilsequences, which have played an important role in recent developments in ergodic theory and additive combinatorics; we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally, meaning on finite intervals. The second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system. We show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1.