scholarly journals EVENTUALLY NUMBER-CONSERVING CELLULAR AUTOMATA

2007 ◽  
Vol 18 (01) ◽  
pp. 35-42 ◽  
Author(s):  
NINO BOCCARA
Keyword(s):  
Cellular Automaton ◽  
Active Sites ◽  
Sufficient Conditions ◽  
Limit Set ◽  
Constant Number ◽  
Interacting Particles ◽  
Limit Sets ◽  
One Dimensional ◽  
Preliminary Study ◽  
Conservation Rule

Although it is undecidable whether a one-dimensional cellular automaton obeys a given conservation law over its limit set, it is however possible to obtain sufficient conditions to be satisfied by a one-dimensional cellular automaton to be eventually number-conserving. We present a preliminary study of two-input one-dimensional cellular automaton rules called eventually number-conserving cellular automaton rules whose limit sets, reached after a number of time steps of the order of the cellular automaton size, consist of states having a constant number of active sites. In particular, we show how to find rules having given limit sets satisfying a conservation rule. Viewed as models of systems of interacting particles, these rules obey a kind of Darwinian principle by either annihilating unnecessary particles or creating necessary ones.

On the sofic limit sets of cellular automata

1995 ◽  
Vol 15 (4) ◽  
pp. 663-684 ◽  
Author(s):  
Alejandro Maass
Keyword(s):  
Fixed Point ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Finite Type ◽  
Finite Time ◽  
Limit Set ◽  
Limit Sets ◽  
One Dimensional ◽  
Sofic System

AbstractIt is not known in general whether any mixing sofic system is the limit set of some one-dimensional cellular automaton. We address two aspects of this question. We prove first that any mixing almost of finite type (AFT) sofic system with a receptive fixed point is the limit set of a cellular automaton, under which it is attained in finite time. The AFT condition is not necessary: we also give examples of non-AFT sofic systems having the same properties. Finally, we show that near Markov sofic systems (a subclass of AFT sofic systems) cannot be obtained as limit sets of cellular automata at infinity.

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

scholarly journals Limit sets of stable cellular automata

2013 ◽  
Vol 35 (3) ◽  
pp. 673-690 ◽  
Author(s):  
ALEXIS BALLIER
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Point Of View ◽  
Limit Set ◽  
Limit Sets ◽  
Sofic Shift ◽  
Almost Everywhere ◽  
Sofic Shifts ◽  
Full Shift ◽  
Factor Map

AbstractWe study limit sets of stable cellular automata from a symbolic dynamics point of view, where they are a special case of sofic shifts admitting a steady epimorphism. We prove that there exists a right-closing almost-everywhere steady factor map from one irreducible sofic shift onto another one if and only if there exists such a map from the domain onto the minimal right-resolving cover of the image. We define right-continuing almost-everywhere steady maps, and prove that there exists such a steady map between two sofic shifts if and only if there exists a factor map from the domain onto the minimal right-resolving cover of the image. To translate this into terms of cellular automata, a sofic shift can be the limit set of a stable cellular automaton with a right-closing almost-everywhere dynamics onto its limit set if and only if it is the factor of a full shift and there exists a right-closing almost-everywhere factor map from the sofic shift onto its minimal right-resolving cover. A sofic shift can be the limit set of a stable cellular automaton reaching its limit set with a right-continuing almost-everywhere factor map if and only if it is the factor of a full shift and there exists a factor map from the sofic shift onto its minimal right-resolving cover. Finally, as a consequence of the previous results, we provide a characterization of the almost of finite type shifts (AFT) in terms of a property of steady maps that have them as range.

scholarly journals A non-Borel special alpha-limit set in the square

2021 ◽  
pp. 1-11
Author(s):  
STEPHEN JACKSON ◽  
BILL MANCE ◽  
SAMUEL ROTH
Keyword(s):  
Dynamical Systems ◽  
Limit Set ◽  
Limit Sets ◽  
Unit Square

Abstract We consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

On Approximate Large Deviations for 1D Diffusion

2003 ◽  
Vol 10 (2) ◽  
pp. 381-399
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Approximation Scheme ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Euler Approximation ◽  
One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

GENERALIZED DECIMATION INVARIANT SEQUENCES IN DIMENSION N

2002 ◽  
Vol 12 (04) ◽  
pp. 709-737 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
One Dimensional ◽  
Necessary And Sufficient

We generalize the concept of one-dimensional decimation invariant sequences, i.e. sequences which are invariant under a specific rescaling, to dimension N. After discussing the elementary properties of decimation-invariant sequences, we focus our interest on their periodicity. Necessary and sufficient conditions for the existence of periodic decimation invariant sequences are presented.

Sign in / Sign up

Export Citation Format

Share Document