Topological entropy and equivalence of dynamical systems

Adler, Roy L.; Marcus, Brian

doi:10.1090/memo/0219

Cited by 144 publications

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“…Let a 1 denote the path from p to the branch point, a 2 the path from the branch point to p, and = a 2 ba 1 …”

Section: Example 410 Suppose That (Y G) Is Given Bymentioning

confidence: 99%

“…The canonicalness requires the dependence on O, and perhaps this is why it was not noticed earlier. However, there have been other works ( [1,11]) achieving some specialness or invariant for SFT covers of some systems, and there were earlier constructions of canonical covers for some systems. Krieger ([8]) gave canonical SFT covers of sofic systems (which are zero-dimensional), and Fried ( [7]) more generally offered canonical coordinate (CC) covers of finitely presented (FP) systems.…”

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confidence: 99%

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Canonical symbolic dynamics for one-dimensional generalized solenoids

2001

Trans. Amer. Math. Soc.

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“…Let a 1 denote the path from p to the branch point, a 2 the path from the branch point to p, and = a 2 ba 1 …”

Section: Example 410 Suppose That (Y G) Is Given Bymentioning

confidence: 99%

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confidence: 99%

Canonical symbolic dynamics for one-dimensional generalized solenoids

2001

Trans. Amer. Math. Soc.

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“…This implies that there exists g ∈ G such that T 1 + g = T 2 . We say that T 2 is a G-translate of T 1 . This does not mean that if T 1 , T 2 ∈ T and g ∈ G such that T 1 + g = T 2 , then T 1 and T 2 are of the same type.…”

Section: Definitionmentioning

confidence: 99%

“…Once we have a Markov partition we will be able to represent the dynamical system (R n mod Z n , φ mod Z n ) as a shift of finite type. In [1] Adler and Marcus outline the essential requirements for a symbolic system to represent an abstract dynamical system. In this final section we review these ideas .…”

Section: Symbolic Representationsmentioning

confidence: 99%

Numeration systems and Markov partitions from self similar tilings

Praggastis

1999

Trans. Amer. Math. Soc.

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Abstract. Using self similar tilings we represent the elements of R n as digit expansions with digits in R n being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms.Fractals and fractal tilings have captured the imaginations of a wide spectrum of disciplines. Computer generated images of fractal sets are displayed in public science centers, museums, and on the covers of scientific journals. Fractal tilings which have interesting properties are finding applications in many areas of mathematics. For example, number theorists have linked fractal tilings of R 2 with numeration systems for R 2 in complex bases [16], [8]. We will see that fractal self similar tilings of R n provide natural building blocks for numeration systems of R n . These numeration systems generalize the 1-dimensional cases in [14], [10], [11] as well as the 2-dimensional cases mentioned above. Our motivation for studying fractal tilings comes from ergodic theory.In [2] Adler and Weiss show that topological entropy is a complete invariant for metric equivalence of continuous ergodic automorphisms of the two-dimensional torus. Their method of proof is to construct a partition of the 2-torus which satisfies certain properties. The partition is called a Markov partition. By assigning each element of the partition a symbol, it is possible to assign each point in the 2-torus a bi-infinite sequence of symbols which corresponds to the orbit of the point. The objective is to represent the continuous dynamical system as a symbolic one in such a way that periodicity and transitivity is preserved in the representation.In [4] Bowen shows that every Anosov diffeomorphism has a Markov partition. In particular every hyperbolic toral automorphism has a Markov partition. His construction uses a recursive definition which deforms rectangles in the stable and unstable directions. While existence is shown, the proof does not indicate an efficient way to actually construct the partitions. In [5] Bowen shows that in the case of the 3-torus the boundary sets of the Markov partition have fractional Hausdorff dimension. In [3] Bedford constructs examples of Markov partitions for the 3-torus and describes the sets as crinkly tin cans. In [6] Cawley generalizes Bowen's results to higher dimensional tori.Since Markov partitions have properties which resemble those of self similar tilings and the boundary sets of these partitions have fractional dimension, it is

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“…of the entries of w (n) is 1, and w\ n) = 0. Then by iteration of (3.3), w (n) is contained in an integral basis {fc (1) ,..., b…”

Section: Examplesmentioning

confidence: 99%

Shift equivalence and the Jordan form away from zero

Boyle

1984

Ergod. Th. Dynam. Sys.

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Abstract. Only finitely many shift equivalence classes of non-negative aperiodic integral matrices may share a given diagonal Jordan form away from zero. The diagonal assumption is necessary. IntroductionSubshifts of finite type are fundamental in applications of symbolic dynamics to diffeomorphisms [3], ergodic theory [4], and coding theory [2]. These subshifts are defined by square non-negative integral matrices [16], and may be studied by way of invariants of matrices which define topologically conjugate subshifts. We will refer to such invariants as 'invariants of conjugacy'.In a fundamental paper [16], Williams introduced two equivalence relations on square non-negative integral matrices: shift equivalence and strong shift equivalence. Strong shift equivalence is a complete but highly non-computable invariant of conjugacy. Shift equivalence is an invariant of conjugacy which is conjectured to be complete and which is more computable-it is often practical to decide if two matrices are shift equivalent, and significant partial results suggest there is a general decision procedure [7]. In addition, from Krieger's work we find shift equivalence intimately related to dimension groups [9] and the construction of factor maps [10].The Jordan form away from zero is a strictly weaker invariant of conjugacy. It is still a very strong one; it determines the entropy and zeta function, classifies irreducible subshifts of finite type up to 'stable weak isomorphism ' [10] and severely constrains equal-entropy factors [8]. Moreover, it is completely computable. To understand the structure of the class of subshifts of finite type, we should in particular understand how shift equivalence refines this Jordan equivalence relation.In this paper, we find that only finitely many shift equivalence classes of aperiodic matrices may share a given diagonal Jordan form away from zero. On the other hand, examples are provided in which infinitely many aperiodic non-negative integral matrices, pairwise not shift equivalent, share the same (non-diagonal) Jordan form away from zero.

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Topological entropy and equivalence of dynamical systems

Cited by 144 publications

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Canonical symbolic dynamics for one-dimensional generalized solenoids

Canonical symbolic dynamics for one-dimensional generalized solenoids

Numeration systems and Markov partitions from self similar tilings

Shift equivalence and the Jordan form away from zero

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