A classification of inverse limit spaces of tent maps with a nonrecurrent critical point

Raines, Brian E.; Štimac, Sonja

doi:10.2140/agt.2009.9.1049

Cited by 13 publications

(20 citation statements)

References 24 publications

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“…Letx i andx i+1 be adjacent q-points in C withx i ≤z <x i+1 . Then, without loss of generality, [14], σ −b (x i ) and h(x i ) are on the same arc-component of a link of C p,n , and the same is true for σ −b (x i+1 ) and h(x i+1 ). Let a = min{σ −b (x i ), h(x i )} and letb = max{σ −1 (x i+1 ), h(x i+1 )} Let B be the arc with endpointsā andb.…”

Section: Definitions and Preliminary Lemmasmentioning

confidence: 86%

“…Let S ∈ N be large enough to satisfy the conditions from [14]. These conditions are quite technical and will mostly not be important in this paper.…”

Section: Definitions and Preliminary Lemmasmentioning

confidence: 99%

“…We write C ≺ D if the chaining C refines the chaining D, and we define the mesh of C to be the largest diameter of any of its links. In [14], we construct a sequence of chainings of K, {C k,r } k∈Z + ,r≥S . Let k ∈ Z + and r ≥ S. Let P be the partition of [0, T (1/2)] induced by the collection of points…”

Section: Definitions and Preliminary Lemmasmentioning

confidence: 99%

See 2 more Smart Citations

Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point

Block¹,

Keesling²,

Raines³

et al. 2009

Topology and its Applications

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Section: Definitions and Preliminary Lemmasmentioning

confidence: 86%

“…Let S ∈ N be large enough to satisfy the conditions from [14]. These conditions are quite technical and will mostly not be important in this paper.…”

Section: Definitions and Preliminary Lemmasmentioning

confidence: 99%

See 1 more Smart Citation

Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point

Block¹,

Keesling²,

Raines³

et al. 2009

Topology and its Applications

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“…Furthermore, they have been the subject of many research articles [12,17,28,30], often in relation to their ω-limit sets [6,21,22,23,24]. In this paper we make several important observations about the behaviour of tent maps, allowing us to prove new results about the nature of their limit sets in relation to certain well-known dynamical properties.…”

Section: Introductionmentioning

confidence: 95%

On the ω-limit sets of tent maps

2012

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Abstract. For a continuous map f on a compact metric space (X, d), a set D ⊂ X is internally chain transitive if for every x, y ∈ D and every δ > 0 there is a sequence of points x = x 0 , x 1 , . . . , x n = y such that d(f (x i ), x i+1 ) < δ for 0 ≤ i < n. It is known that every ω-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed set D ⊂ X is internally chain transitive if and only if D is an ω-limit set for some point in X, and that the same is also true for the full tent map T 2 : [0, 1] → [0, 1]. In this paper, we prove that for tent maps with periodic critical point every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an ω-limit set. Together, these results lead us to conjecture that for maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.

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“…The key observation in our proof is Proposition 3.7 which implies that every homeomorphism h maps symmetric arcs to symmetric arcs, not just to quasi-symmetric arcs. (The difficulty that quasi-symmetric arcs pose was first observed and overcome in [22] in the setting of tent maps with non-recurrent critical point.) To prove Proposition 3.7, the special structure of the Fibonacci-like maps, and especially the special chains it allows, is used.…”

Section: Introductionmentioning

confidence: 99%

Fibonacci-like unimodal inverse limit spaces and the core Ingram conjecture

Bruin¹,

Štimac²

2015

TMNA

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We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allow us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to the proof of the Ingram Conjecture for cores of Fibonacci-like unimodal inverse limits.2000 Mathematics Subject Classification. 54H20, 37B45, 37E05. Key words and phrases. tent map, inverse limit space, Fibonacci unimodal map, structure of inverse limit spaces.It is this version that we solve here for Fibonacci-like tent maps: Main Theorem 1. If 1 ≤ s < s ′ ≤ 2 are the parameters of Fibonacci-like tent-maps, then the corresponding cores of inverse limit spaces lim ← − ([c 2 , c 1 ], T s ) and lim ← − ([c 2 , c 1 ], T s ′ ) are non-homeomorphic. The core version of the Ingram Conjecture was proved already in the postcritically finite case, since neither Kailhofer [17], norŠtimac [24] use the arc-component C, but work on some other arc-components of the core (although not on the arc-component of the fixed point (. . . , r, r, r) which we use in this paper).

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A classification of inverse limit spaces of tent maps with a nonrecurrent critical point

Abstract: In this paper we prove the nonrecurrent case of the Ingram conjecture by showing that if T s and T t are two tent maps with nonrecurrent critical points then lim fOE0; 1; T s g is homeomorphic to lim fOE0; 1; T t g if and only if s D t . 37B10; 37B45

Cited by 13 publications

References 24 publications

Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point

Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point

On the ω-limit sets of tent maps

Fibonacci-like unimodal inverse limit spaces and the core Ingram conjecture

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