scholarly journals Coding of substitution dynamical systems as shifts of finite type

2014 ◽  
Vol 36 (3) ◽  
pp. 944-972 ◽  
Author(s):  
PAUL SURER
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Periodic Points ◽  
Basic Properties ◽  
Substitution Dynamical Systems ◽  
Shift Map ◽  
Special Case

We develop a theory that allows us to code dynamical systems induced by primitive substitutions continuously as shifts of finite type in many different ways. The well-known prefix–suffix coding turns out to correspond to one special case. We precisely analyse the basic properties of these codings (injectivity, coding of the periodic points, properties of the presentation graph, interaction with the shift map). A lot of examples illustrate the theory and show that, depending on the particular coding, several amazing effects may occur. The results give new insights into the theory of substitution dynamical systems and might serve as a powerful tool for further researches.

scholarly journals Local escape rates for -mixing dynamical systems

2019 ◽  
Vol 40 (10) ◽  
pp. 2854-2880
Author(s):  
N. HAYDN ◽  
F. YANG
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Periodic Points ◽  
Extremal Index ◽  
Equilibrium States ◽  
Gibbs States ◽  
Markov Systems ◽  
Interval Maps ◽  
Subshifts Of Finite Type ◽  
Young Towers

We show that dynamical systems with $\unicode[STIX]{x1D719}$-mixing measures have local escape rates which are exponential with rate 1 at non-periodic points and equal to the extremal index at periodic points. We apply this result to equilibrium states on subshifts of finite type, Gibbs–Markov systems, expanding interval maps, Gibbs states on conformal repellers, and more generally to Young towers, and by extension to all systems that can be modeled by a Young tower.

scholarly journals Periodic Intermediate β-Expansions of Pisot Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 903
Author(s):  
Blaine Quackenbush ◽  
Tony Samuel ◽  
Matt West
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Markov Property ◽  
Periodic Points ◽  
Pisot Number ◽  
Subshift Of Finite Type ◽  
Pisot Numbers ◽  
Type Property ◽  
Perron Number ◽  
Dense Set

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are β -shifts, namely transformations of the form T β , α : x ↦ β x + α mod 1 acting on [ − α / ( β − 1 ) , ( 1 − α ) / ( β − 1 ) ] , where ( β , α ) ∈ Δ is fixed and where Δ ≔ { ( β , α ) ∈ R 2 : β ∈ ( 1 , 2 ) and 0 ≤ α ≤ 2 − β } . Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045–2055, 2019), that the set of ( β , α ) such that T β , α has the subshift of finite type property is dense in the parameter space Δ . Here, they proposed the following question. Given a fixed β ∈ ( 1 , 2 ) which is the n-th root of a Perron number, does there exists a dense set of α in the fiber { β } × ( 0 , 2 − β ) , so that T β , α has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269–278, 1980) from the case when α = 0 to the case when α ∈ ( 0 , 2 − β ) . That is, we examine the structure of the set of eventually periodic points of T β , α when β is a Pisot number and when β is the n-th root of a Pisot number.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

CNN: A Vision of Complexity

1997 ◽  
Vol 07 (10) ◽  
pp. 2219-2425 ◽  
Author(s):  
Leon O. Chua
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Real Numbers ◽  
Initial State ◽  
Brain Science ◽  
Von Neumann ◽  
Local Activity ◽  
Sphere Of Influence ◽  
Nonlinear Pde’S ◽  
Special Case

CNN is an acronym for either Cellular Neural Network when used in the context of brain science, or Cellular Nonlinear Network when used in the context of coupled dynamical systems. A CNN is defined by two mathematical constructs: 1. A spatially discrete collection of continuous nonlinear dynamical systems called cells, where information can be encrypted into each cell via three independent variables called input, threshold, and initial state. 2. A coupling law relating one or more relevant variables of each cell Cij to all neighbor cells Ckl located within a prescribed sphere of influence Sij(r) of radius r, centered at Cij. In the special case where the CNN consists of a homogeneous array, and where its cells have no inputs, no thresholds, and no outputs, and where the sphere of influence extends only to the nearest neighbors (i.e. r = 1), the CNN reduces to the familiar concept of a nonlinear lattice. The bulk of this three-part exposition is devoted to the standard CNN equation [Formula: see text] where xij, yij, uij and zij are scalars called state, output, input, and threshold of cell Cij; akl and bkl are scalars called synaptic weights, and Sij(r) is the sphere of influence of radius r. In the special case where r = 1, a standard CNN is uniquely defined by a string of "19" real numbers (a uniform thresholdzkl = z, nine feedback synaptic weights akl, and nine control synaptic weights bkl) called a CNN gene because it completely determines the properties of the CNN. The universe of all CNN genes is called the CNN genome. Many applications from image processing, pattern recognition, and brain science can be easily implemented by a CNN "program" defined by a string of CNN genes called a CNN chromosome. The first new result presented in this exposition asserts that every Boolean function of the neighboring-cell inputs can be explicitly synthesized by a CNN chromosome. This general theorem implies that every cellular automata (with binary states) is a CNN chromosome. In particular, a constructive proof is given which shows that the game-of-life cellular automata can be realized by a CNN chromosome made of only three CNN genes. Consequently, this "game-of-life" CNN chromosome is a universal Turing machine, and is capable of self-replication in the Von Neumann sense [Berlekamp et al., 1982]. One of the new concepts presented in this exposition is that of a generalized cellular automata (GCA), which is outside the framework of classic cellular (Von Neumann) automata because it cannot be defined by local rules: It is simply defined by iterating a CNN gene, or chromosome, in a "CNN DO LOOP". This new class of generalized cellular automata includes not only global Boolean maps, but also continuum-state cellular automata where the initial state configuration and its iterates are real numbers, not just a finite number of states as in classical (von Neumann) cellular automata. Another new result reported in this exposition is the successful implementation of an analog input analog output CNN universal machine, called a CNN universal chip, on a single silicon chip. This chip is a complete dynamic array stored-program computer where a CNN chromosome (i.e. a CNN algorithm or flow chart) can be programmed and executed on the chip at an extremely high speed of 1 Tera (1012) analog instructions per second (based on a 100 × 100 chip). The CNN universal chip is based entirely on nonlinear dynamics and therefore differs from a digital computer in its fundamental operating principles. Part II of this exposition is devoted to the important subclass of autonomous CNNs where the cells have no inputs. This class of CNNs can exhibit a great variety of complex phenomena, including pattern formation, Turing patterns, knots, auto waves, spiral waves, scroll waves, and spatiotemporal chaos. It provides a unified paradigm for complexity, as well as an alternative paradigm for simulating nonlinear partial differential equations (PDE's). In this context, rather than regarding the autonomous CNN as an approximation of nonlinear PDE's, we advocate the more provocative point of view that nonlinear PDE's are merely idealizations of CNNs, because while nonlinear PDE's can be regarded as a limiting form of autonomous CNNs, only a small class of CNNs has a limiting PDE representation. Part III of this exposition is rather short but no less significant. It contains in fact the potentially most important original results of this exposition. In particular, it asserts that all of the phenomena described in the complexity literature under various names and headings (e.g. synergetics, dissipative structures, self-organization, cooperative and competitive phenomena, far-from-thermodynamic equilibrium phenomena, edge of chaos, etc.) are merely qualitative manifestations of a more fundamental and quantitative principle called the local activity dogma. It is quantitative in the sense that it not only has a precise definition but can also be explicitly tested by computing whether a certain explicitly defined expression derived from the CNN paradigm can assume a negative value or not. Stated in words, the local activity dogma asserts that in order for a system or model to exhibit any form of complexity, such as those cited above, the associated CNN parameters must be chosen so that either the cells or their couplings are locally active.

Simultaneous Resonance and Anti-Resonance in Dynamical Systems Under Asynchronous Parametric Excitation

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Artem Karev ◽  
Peter Hagedorn
Keyword(s):  
Dynamical Systems ◽  
Parametric Excitation ◽  
Analytical Method ◽  
Normal Forms ◽  
High Sensitivity ◽  
Combination Resonance ◽  
Concurrent Study ◽  
The Stability ◽  
Analytical Expressions ◽  
Special Case

Abstract Since the discovery of parametric anti-resonance, parametric excitation has also become more prominent for its stabilizing properties. While resonance and anti-resonance are mostly studied individually, there are systems where both effects appear simultaneously at each combination resonance frequency. With a steep transition between them and a high sensitivity of their relative positions, there is a need for a concurrent study of resonance and anti-resonance. The semi-analytical method of normal forms is used to derive approximate analytical expressions describing the magnitude of the stability impact as well as the precise locations of stabilized and destabilized areas. The results reveal that the separate appearance of resonance and anti-resonance is only a special case occurring for synchronous parametric excitation. In particular, in circulatory systems the simultaneous appearance is expected to be much more common.

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