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- Title
Topological Expansive Lorenz Maps with a Hole at Critical Point.
- Authors
Sun, Yun; Li, Bing; Ding, Yiming
- Abstract
Let f be an expansive Lorenz map and c be the critical point. The survivor set is denoted as S f (H) : = { x ∈ [ 0 , 1 ] : f n (x) ∉ H , ∀ n ≥ 0 } , where H is an open subinterval. Here we study the hole H = (a , b) with a ≤ c ≤ b and a ≠ b . We observe that the case a = c is equivalent to the hole at 0, and the case b = c is equivalent to the hole at 1. Given any expansive Lorenz map f with a hole H = (a , b) and S f (H) ⫅̸ { 0 , 1 } , we prove that there exists a Lorenz map g such that S ~ f (H) \ Ω (g) is countable, where Ω (g) is the Lorenz-shift of g and S ~ f (H) is the symbolic representation of S f (H) . Moreover, let a be fixed, we also give a complete characterization of the maximal plateau I(b) such that for all ϵ ∈ I (b) , S f + (a , ϵ) = S f + (a , b) , and I(b) may degenerate to a single point b. As an application, when f has an ergodic acim and a is fixed, we obtain that the topological entropy function λ f (a) : b ↦ h top (f | S f (a , b)) is a devil staircase. At the special case that f being an intermediate β -transformation, using the Ledrappier-Young formula, the Hausdorff dimension function η f (a) : b ↦ dim H (S f (a , b)) is naturally a devil staircase when fixing a. All the results can be naturally extended to the case that b is fixed. As a result, we extend the devil staircases in (Kalle et al. in Ergod Th Dyn Syst 40:2482–2514, 2020; Langeveld and Samuel in Acta Math Hungar 170:269–301, 2023; Urbanski in Ergod Th Dyn Syst 6:295–309, 1986) to expansive Lorenz maps with a hole at critical point.
- Subjects
TOPOLOGICAL entropy; FRACTAL dimensions; LORENZ equations; STAIRCASES; CRITICAL point theory
- Publication
Journal of Statistical Physics, 2024, Vol 191, Issue 5, p1
- ISSN
0022-4715
- Publication type
Article
- DOI
10.1007/s10955-024-03265-0