Specification and Towers in Shift Spaces

Climenhaga, Vaughn

doi:10.1007/s00220-018-3265-y

Cited by 18 publications

(30 citation statements)

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“…It was shown by the second author in [11] that almost specification does not imply intrinsic ergodicity, even for g ≡ 4. To prove Theorem 1.1, we will show that left almost specification with any constant mistake function implies a version of the non-uniform specification property from [5,6]. This property requires the existence of C p , G, C s ⊂ L(X) such that…”

Section: Proofmentioning

confidence: 98%

“…These conditions appeared in [6] (in a mildly different form), where a third condition was also required that controls how quickly words of the form uvw with u ∈ C p , v ∈ G, w ∈ C s with |u|, |w| ≤ M can be extended to words in G. In our setting of LAS with bounded g, we produce C p , G, C s satisfying [I]-[II], but it is not clear whether the collections we produce satisfy this third condition, and so we cannot apply the results from [6]. Rather, we use the following conditions on G that were introduced in [5], which we are able to verify in our setting; roughly speaking, these ask that intersections and unions of words in G are again in G (under some mild conditions).…”

Section: Proofmentioning

confidence: 99%

“…A uniqueness result with non-uniform specification. To prove Theorem 1.1, we will need the following uniqueness result, which combines several theorems and remarks from [5]; see §3.4 for details. .…”

Section: 5mentioning

confidence: 99%

“…It is clear that X β,β ′ is a factor of the direct product X β × X β ′ (induced by the letter-to-letter map (i, j) → i + j), and so every X β,β ′ also has a unique MME. It is quite possible that the methods of [6] or [5] could also be used to prove uniqueness of MME for these shifts, but it would be more difficult.…”

Section: Introductionmentioning

confidence: 99%

“…Is there a subshift X with LAS for some sublinear mistake function for which there are multiple fully supported ergodic MMEs?The mechanism for uniqueness in Theorem 1.1 is a result from [5] that uses a certain non-uniform specification condition (superficially quite different from almost specification) to model X with a countable state Markov shift. The conditions in [5] are mild variants of conditions introduced in [6], where they were used to prove that not only are β-shifts intrinsically ergodic, but so are their subshift factors. Although the conditions from [6] are well-behaved under passing to factors, their behaviour upon passing to products is not immediately clear.…”

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confidence: 99%

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One-sided almost specification and intrinsic ergodicity

Climenhaga¹,

Pavlov²

2018

Ergod. Th. Dynam. Sys.

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We define a new property called one-sided almost specification, which lies between the properties of specification and almost specification, and prove that it guarantees intrinsic ergodicity (i.e. uniqueness of the measure of maximal entropy) if the corresponding mistake function g is bounded. We also show that uniqueness may fail for unbounded g such as log log n. Our results have consequences for almost specification: we prove that almost specification with g ≡ 1 implies one-sided almost specification (with g ≡ 1), and hence uniqueness. On the other hand, the second author showed recently that almost specification with g ≡ 4 does not imply uniqueness.1 Recently a number of weakened versions of the specification property have been used to study various questions in ergodic theory. This includes almost specification [8,11,13,14,18], which allows one to concatenate arbitrary words in the language into a new word in the language if a "small" number of letters are allowed to change in each word. The number of changes is controlled by a sublinear "mistake function" g(n); see Section 2 for a formal definition. Almost specification is enough to establish various results in large deviation theory [14] and multifractal analysis [13,18], but it was unknown for some time whether almost specification also implies intrinsic ergodicity. This question was answered in the negative by (independently) [9] and [11]; in fact, [11, Theorem 1.2] shows that intrinsic ergodicity may fail even with the constant mistake function g ≡ 4. (Here and elsewhere, g ≡ C means that g is the constant function C.)One motivation for almost specification is the fact that many natural examples satisfy it for small g, such as the classical β-shifts, which have almost specification with g ≡ 1 and do satisfy intrinsic ergodicity. In fact, β-shifts satisfy a slightly stronger property; when one wishes to concatenate some words w (1) , . . ., w (n) together, it suffices to make the permitted number of changes to the words w (1) , . . . , w (n−1) , and leave the final word w (n) untouched. Though this might seem like an incremental strengthening, it proves quite important. We call this stronger notion left almost specification (LAS); our main result is that this property actually does imply intrinsic ergodicity if the mistake function g is bounded.Theorem 1.1. If X is a subshift with left almost specification for a bounded mistake function, then it has a unique measure of maximal entropy µ. Moreover, µ is the limiting distribution of periodic orbits; finally, the system (X, σ, µ) is Bernoulli, and has exponential decay of correlations and the central limit theorem for Hölder observables.We prove Theorem 1.1 in §3. This theorem covers the case when X has the usual specification property (see Lemma 2.15). We show in §4.1 that it also covers the case when X has the usual almost specification property with g ≡ 1, and deduce the following.Corollary 1.2. If X is a subshift with almost specification for the mistake function g ≡ 1, then it has a unique measure of ...

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Section: Proofmentioning

confidence: 98%