Critical itineraries of maps with constant slope and one discontinuity

Barnsley, Michael F.; Steiner, Wolfgang; Vince, Andrew

doi:10.1017/s0305004114000486

Cited by 8 publications

(5 citation statements)

References 30 publications

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“…Since n can be chosen arbitrarily large, then (k + , k − ) can be arbitrarily close to (v, w) with respect to the usual metric. By construction we know that (k + , k − ) and (v, w) have the same numerator of kneading determinants, hence they have the same β and (k + , k − ) satisfies definition 2.2 (3). By the proof of lemma 3.1, we have (v, w) is prime if and only if (k + , k − ) is prime.…”

Section: Proof Of Main Resultsmentioning

confidence: 89%

“…where (k + , k − ) is the pair of sequence obtained by replacing 1's by w + , replacing 0's by w − in k 1 + and k 1 − . Using * -product, (1) and ( 2) can be expressed by (3). So the kneading invariants are renormalizable if and only if it can be decomposed as the * -product; otherwise, we say (k + , k − ) is prime.…”

Section: Combinatorial Renormalizationmentioning

confidence: 99%

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Fiber denseness of intermediate β-shifts of finite type

Sun,

Li,

Ding

2023

Nonlinearity

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We focus on T β , α ( x ) = β x + α ( m o d 1), x ∈ [ 0 , 1 ] and ( β , α ) ∈ Δ := { ( β , α ) ∈ R 2 : β ∈ ( 1 , 2 ) and 0 < α < 2 − β } . The T β , α ± -expansions τ β , α ± ( x ) of critical point c β , α = 1 − α β are called kneading invariants, denoted as ( k + , k − ) . Let Δ ( k + ) := { ( β , α ) ∈ Δ : τ β , α + ( c β , α ) = k + } with k + being periodic, we state that Δ ( k + ) is a smooth curve which can be regarded as a fiber. By combinatorial method, we extend the results of Parry (1960 Acta Math. Acad. Sci. Hung. 11 401–16) and show that, the set of ( β , α ) with its Ω β , α being a SFT is dense in Δ ( k + ) . Similarly for the fiber Δ ( k − ) . When considering another fiber Δ ( β ) := { ( β , α ) ∈ Δ : β ∈ ( 1 , 2 ) is fixed } , we demonstrate that when β is not a multinacci number, there are only countably many distinct matching intervals on Δ ( β ) . Using Markov approximation, we prove that the set of ( β , α ) with Ω β , α being a SFT is dense in each matching interval. We also propose a classification scheme for the endpoints of these matching intervals.

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Section: Proof Of Main Resultsmentioning

confidence: 89%

Section: Combinatorial Renormalizationmentioning

confidence: 99%

Fiber denseness of intermediate β-shifts of finite type

Sun,

Li,

Ding

2023

Nonlinearity

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“…which means the first l words of k − and the first r words of k + , then we have (1) e − = w ∞ − and e + = w ∞ + , i.e., e − and e + are periodic; (2) The following decompositions holds:…”

Section: Renormalization Of Expansive Lorenz Mapsmentioning

confidence: 99%

“…In 1996, Paul Glendinning and Toby Hall [7] proved a similar result for Lorenz maps. In 2014, work directly with the symbolic space and do not require it to be the address space of some map, Barnsley [2] put forward the following result.…”

Section: Calculation Of α and βmentioning

confidence: 99%

Complete invariants and parametrization of expansive Lorenz maps

Ding,

Sun

2021

Preprint

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“…The entropy formula was introduced in Glendinning and Hall [15] in the context of for Lorenz maps. Recently, Barnsley, Steiner and Vince in [4] give a symbolic proof showing that the smallest positive root of K(t), denoted by κ, satisfies…”

Section: Introductionmentioning

confidence: 99%