A generalized shadowing lemma

2002/01/01 by Shaobo Gan · 1 voice · 3 citations
Mathematics · #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology #Analytic and geometric function theory

paper · doi:10.3934/dcds.2002.8.627

openalex publication_date 2002/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/17

Abstract

In this paper, we prove a generalized shadowing lemma. Let f ∈ Diff(M).Assume that Λ is a closed invariant set of f and there is a continuous invariantsplitting TΛ M = E⊕ F on Λ. For any λ ∈ (0, 1) there exist L > 0, d₀> 0 suchthat for any d ∈ (0, d₀] and any λ-quasi-hyperbolic d-pseudoorbit \x_i, n_i\_i=-∞^∞,there exists a point x which Ld-shadows \x_i, n_i\_i=-∞^∞. Moreover, if \x_i, n_i\_i=-∞^∞ is periodic, i.e., there exists an m > 0 such that x_i+m= x_i and n_i+m = n_i for all i,then the point x can be chosen to be periodic.

Cited by

• Ledrappier-Young entropy formula for C¹ diffeomorphisms with dominated splitting Part 1: Unstable entropy formula and invariance principle
• Diffeomorphisms with Liao-Pesin set
• Hyperbolic Periodic Points and Hyperbolic Measures with Dominated Splitting

Discussions

• Rigorously speaking, that's not exactly what the lemma says: It says that there is a (exact) trajectory arbitrarily close to the numerical one, it doesn't guarantee they're identical. Also, notice tha [stackexchange, 1 points]

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