Beta-Shifts, Their Languages, and Computability

Simonsen, Jakob Grue

doi:10.1007/s00224-009-9245-z

Cited by 3 publications

(8 citation statements)

References 37 publications

(33 reference statements)

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“…Beta-shifts are symbolic systems that were introduced by Rényi in [54]. See [47,63] for subsequent developments and additional references. Beta-shifts have been studied both from the computability and the number theoretical point of view.…”

Section: Beta-shiftsmentioning

confidence: 99%

Computability of topological pressure on compact shift spaces beyond finite type*

et al. 2022

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We investigate the computability (in the sense of computable analysis) of the topological pressure P top(ϕ) on compact shift spaces X for continuous potentials ϕ : X → R . This question has recently been studied for subshifts of finite type (SFTs) and their factors (sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalised gap shifts, and particular beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalised pressure function (X, ϕ) ↦ P top(X, ϕ| X ) is not computable for a large set of shift spaces X and potentials ϕ. In particular, the entropy map X ↦ h top(X) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.

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Section: Beta-shiftsmentioning

confidence: 99%

Computability of topological pressure on compact shift spaces beyond finite type*

et al. 2022

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“…In particular, Beta-shifts have been studied both from the computability as well as the number theoretical point of view. We refer the reader to [14,17,34,57] for more details about the properties of Beta-shifts. We briefly recall the basic properties of Beta-shifts following [14,17,57].…”

Section: Computability Of the Topological Pressure Of Particular Subs...mentioning

confidence: 99%

“…We refer the reader to [14,17,34,57] for more details about the properties of Beta-shifts. We briefly recall the basic properties of Beta-shifts following [14,17,57].…”

Section: Computability Of the Topological Pressure Of Particular Subs...mentioning

confidence: 99%

“…Since all remaining entries in the p-and q-expansions of x consist of characters less than b 1 and there is a unique backward generator for each character less than b 1 , it follows that the p-and q-expansions of x are identical and the shift space is uniquely representable. It is shown in [57] that the language of the Beta-shift is recursive iff β is a computable real number. Moreover, since β is a computable non-preperiodic real number, there exists a Turing machine that implements the greedy algorithm to compute b(β).…”

Section: Computability Of the Topological Pressure Of Particular Subs...mentioning

confidence: 99%

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Computability of topological pressure on compact shift spaces beyond finite type

Burr¹,

Das²,

Wolf³

et al. 2020

Preprint

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We investigate the computability (in the sense of computable analysis) of the topological pressure Ptop(φ) on compact shift spaces X for continuous potentials φ : X → R. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and Beta shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalized pressure function (X, φ) → Ptop(X, φ|X ) is not computable for a large set of shift spaces X and potentials φ. In particular, the entropy map X → htop(X) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.

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“…This language is recursive (that is, decidable by a Turing machine), if and only if β is a computable real number. [23] The zeta function, and a lap-counting function, are given by Lagarias [24]. The Hausdorff dimension, the topological entropy and general notions of topological pressure arising from conditional variational principles is given by Daniel Thompson [25].…”

Section: Beta Transformation Literature Review and Referencesmentioning

confidence: 99%

On the Beta Transformation

Vepstas

2018

Preprint

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The beta transformation is the iterated map β x mod 1. The special case of β = 2 is known as the Bernoulli map, and is exactly solvable. The Bernoulli map provides a model for pure, unrestrained chaotic (ergodic) behavior: it is the full invariant shift on the Cantor space {0, 1} ω . The Cantor space consists of infinite strings of binary digits; it is notable for many properties, including that it can represent the real number line.The beta transformation defines a subshift: iterated on the unit interval, it singles out a subspace of the Cantor space that is invariant under the action of the left-shift operator. That is, lopping off one bit at a time gives back the same subspace.The beta transform seems to capture something basic about the multiplication of two real numbers: β and x. It offers insight into the nature of multiplication. Iterating on multiplication, one would get β n x -that is, exponentiation; the mod 1 of the beta transform contorts this in strange ways.Analyzing the beta transform is difficult. The work presented here is more-orless a research diary: a pastiche of observations and some shallow insights. One is that chaos seems to be rooted in how the carry bit behaves during multiplication. Another is that one can surgically insert "islands of stability" into chaotic (ergodic) systems, and have some fair amount of control over how those islands of stability behave. One can have islands with, or without a period-doubling "route to chaos".The eigenvalues of the transfer operator seem to lie on a circle of radius 1/β in the complex plane. Given that the transfer operator is purely real, the appearance of such a quasi-unitary spectrum unexpected. The spectrum appears to be the limit of a dense set of quasi-cyclotomic polynomials, the positive real roots of which include the Golden and silver ratios, the Pisot numbers, the n-bonacci (tribonacci, tetranacci, etc.) numbers.

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Beta-Shifts, Their Languages, and Computability

Cited by 3 publications

References 37 publications

Computability of topological pressure on compact shift spaces beyond finite type*

Computability of topological pressure on compact shift spaces beyond finite type*

Computability of topological pressure on compact shift spaces beyond finite type

On the Beta Transformation

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