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- Title
Study on Strong Sensitivity of Systems Satisfying the Large Deviations Theorem.
- Authors
Li, Risong; Lu, Tianxiu; Yang, Xiaofang; Jiang, Yongxi
- Abstract
Let (S , τ) be a nontrivial compact metric space with metric τ and g : S → S be a continuous self-map, ℬ (S) be the sigma-algebra of Borel subsets of S , and ν be a Borel probability measure on (S , ℬ (S)) with ν (V) ∈ (0 , + ∞) for any open subset V ≠ ∅ of S. This paper proves the following results : (1) If the pair (g , ν) has the property that for any β > 0 , there is r (β) > 0 such that ν p ∈ S : 1 m ∑ i = 0 m − 1 V (g i (p)) − ∫ V d ν > β ≤ e − m r (β) , for any open subset V ≠ ∅ of S and all n ≥ 1 sufficiently large (where V is the characteristic function of the set V), then the following hold : (a) The map g is topologically ergodic. (b) The upper density μ ¯ (N g (p , V)) of N g (p , V) is positive for any open subset V ≠ ∅ of S , where N g (p , V) = { m ∈ { 0 , 1 , ... } : g m (p) ∈ V }. (c) There is a g -invariant Borel probability measure ν having full support (i.e. supp (ν) = S). (d) Sensitivity of the map g implies positive lower density sensitivity, hence ergodical sensitivity. (2) If μ ̲ (N g (A , B)) = 1 for any two nonempty open subsets A , B , then there exists λ > 0 satisfying μ ̲ (N g (C , λ)) = 1 for any nonempty open subset C ⊂ S , where N g (C , λ) = { l ∈ { 0 , 1 , ... } : there exist a , b ∈ C with τ (g l (a) , g l (b)) > λ }.
- Subjects
METRIC spaces; PROBABILITY measures; BOREL subsets; LARGE deviations (Mathematics); CHARACTERISTIC functions; SET functions; COMPACT spaces (Topology)
- Publication
International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, 2021, Vol 31, Issue 10, p1
- ISSN
0218-1274
- Publication type
Article
- DOI
10.1142/S0218127421501510