scholarly journals Topological and symbolic dynamics for hyperbolic systems with holes

2010 ◽  
Vol 31 (5) ◽  
pp. 1305-1323 ◽  
Author(s):  
STEFAN BUNDFUSS ◽  
TYLL KRÜGER ◽  
SERGE TROUBETZKOY
Keyword(s):  
Finite Type ◽  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Symbolic Representation ◽  
Invariant Set ◽  
Subshift Of Finite Type ◽  
Axiom A ◽  
Topologically Transitive ◽  
Markov Map

AbstractWe consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Ω* of orbits which never falls into a fixed hole. We study various aspects of the symbolic representation of Ω* and of its non-wandering set Ωnw. Our results are on the cardinality of the set of topologically transitive components of Ωnw and their structure. We also prove that Ω* is generically a subshift of finite type in several senses.

Languages, equicontinuity and attractors in cellular automata

1997 ◽  
Vol 17 (2) ◽  
pp. 417-433 ◽  
Author(s):  
PETR KŮRKA
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
State Space ◽  
Finite Type ◽  
Measure Zero ◽  
Initial Conditions ◽  
The State ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
The Third

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].

THE EFFECTIVE POTENTIAL AND TRANSSHIPMENT IN THERMODYNAMIC FORMALISM AT ZERO TEMPERATURE

2012 ◽  
Vol 13 (01) ◽  
pp. 1250009 ◽  
Author(s):  
EDUARDO GARIBALDI ◽  
ARTUR O. LOPES
Keyword(s):  
Finite Type ◽  
Effective Potential ◽  
Transition Matrix ◽  
Thermodynamic Formalism ◽  
Absolute Temperature ◽  
Zero Temperature ◽  
Hölder Continuous ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
Unique Eigenvalue

For a topologically transitive subshift of finite type defined by a symmetric transition matrix, we introduce a temperature-based problem related to the usual thermodynamic formalism. This problem is described by an operator acting on Hölder continuous observables which is actually superlinear with respect to the max-plus algebra. We thus show that, for each fixed absolute temperature, such an operator admits a unique eigenfunction and a unique eigenvalue. We also study the convergence as the temperature goes to zero and we relate the limit objects to an ergodic version of Kantorovich transshipment problem.

scholarly journals Existence of a measurable saturated compensation function between subshifts and its applications

2010 ◽  
Vol 31 (5) ◽  
pp. 1563-1589 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Symbolic Representation ◽  
Invariant Set ◽  
Conformal Map ◽  
Constant Multiple ◽  
Shift Of Finite Type ◽  
Ergodic Measures ◽  
Borel Measurable ◽  
Compensation Function

AbstractWe show the existence of a bounded Borel measurable saturated compensation function for any factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. These problems were studied in [23] for a compact invariant set whose symbolic representation is a shift of finite type under the condition of the existence of a saturated compensation function. By using the ergodic equilibrium states of a constant multiple of a Borel measurable compensation function, we extend the results to the general case where this condition might not hold, presenting a formula for the Hausdorff dimension for a compact invariant set whose symbolic representation is a subshift and studying invariant ergodic measures of full dimension. We study uniqueness and properties of such measures for a compact invariant set whose symbolic representation is a topologically mixing shift of finite type.

Textile systems and one-sided resolving automorphisms and endomorphisms of the shift

2008 ◽  
Vol 28 (1) ◽  
pp. 167-209 ◽  
Author(s):  
MASAKAZU NASU
Keyword(s):  
Finite Type ◽  
Structure Theory ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
Subshifts Of Finite Type

AbstractTwo results on textile systems are obtained. Using these we prove that for any automorphism φ of any topologically-transitive subshift of finite type, if φ is expansive and φ or φ−1 has memory zero or anticipation zero, then φ is topologically conjugate to a subshift of finite type. Moreover, this is generalized to a result on chain recurrent onto endomorphisms of topologically-transitive subshifts of finite type. Using textile systems and textile subsystems, we develop a structure theory concerning expansiveness with the pseudo orbit tracing property on directionally essentially weakly one-sided resolving automorphisms and endomorphisms of subshifts.

Singular perturbation of symbolic dynamics via thermodynamic formalism

2008 ◽  
Vol 28 (4) ◽  
pp. 1261-1289 ◽  
Author(s):  
TAKEHIKO MORITA ◽  
HARUYOSHI TANAKA
Keyword(s):  
Singular Perturbation ◽  
Finite Type ◽  
Symbolic Dynamics ◽  
Thermodynamic Formalism ◽  
Unperturbed System ◽  
Large Parameter ◽  
Subshift Of Finite Type ◽  
Perturbed System ◽  
Perturbed Systems

AbstractWe consider singular perturbation of a mixing subshift of finite type by means of thermodynamic formalism. In our formulation, the perturbed systems are described by a family of potentials {Φ(α,⋅)} with large parameter α on a fixed subshift of finite type, and the original (unperturbed) system is characterized as the system at infinity obtained by collapsing the perturbed system upon taking $\alpha \to \infty $. We apply our formulation to the collapse of cookie-cutter systems and dispersing open billiards.

Perturbations of multidimensional shifts of finite type

2010 ◽  
Vol 31 (2) ◽  
pp. 483-526 ◽  
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Symbolic Dynamics ◽  
Upper And Lower Bounds ◽  
Shift Of Finite Type ◽  
Strongly Irreducible ◽  
Dimensional Cube

AbstractIn this paper, we study perturbations of multidimensional shifts of finite type. Specifically, for any ℤd shift of finite type X with d>1 and any finite pattern w in the language of X, we denote by Xw the set of elements of X not containing w. For strongly irreducible X and patterns w with shape a d-dimensional cube, we obtain upper and lower bounds on htop (X)−htop (Xw) dependent on the size of w. This extends a result of Lind for d=1 . We also apply our methods to an undecidability question in ℤd symbolic dynamics.

scholarly journals Symbolic dynamics for non-uniformly hyperbolic systems

2020 ◽  
pp. 1-68
Author(s):  
YURI LIMA
Keyword(s):  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Markov Partition ◽  
Equilibrium Measures ◽  
Markov Partitions ◽  
Recent Advances ◽  
Closed Orbits ◽  
Symbolic Extension ◽  
Uniformly Hyperbolic Systems

Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

SPECTRA OF RECURRENCE DIMENSION FOR ADIC SYSTEMS

2002 ◽  
Vol 02 (04) ◽  
pp. 599-607 ◽  
Author(s):  
VÍCTOR F. SIRVENT
Keyword(s):  
Finite Type ◽  
Subshift Of Finite Type ◽  
Poincare Recurrence ◽  
Definition Of ◽  
Poincaré Recurrence

We compute the spectra of the recurrence dimension for adic systems and sub-adic systems. This dimension is characterized by the Poincaré recurrence of the system, and known in the literature as Afraimovich–Pesin dimension. These spectra are invariant under bi-Lipschitz transformations. We show that there is a duality between the spectra of an adic system and the corresponding subshift of finite type. We also consider Billingsley-like definition of the spectra of the recurrence dimension.

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