Lyapunov Exponents of Certain 1D Sensitive Open CA with a Continuum of δ−Periodic Points
Janusz Matyja
Let 𝒜ℤ be a metric Cantor space of bi-infinite words and (𝒜ℤ, σ) its corresponding full shift. Consider surjective CA (𝒜ℤ, F), (𝒜ℤ, σ) with the Borel uniform Bernoulli measure μ. Let ℎμ(𝒜ℤ, 𝐹) and ℎμ(𝒜ℤ, σ) denote KS-entropies of suitable CA. Denote by ℎ(𝒜ℤ, 𝐹) the topological entropy of (𝒜ℤ, 𝐹). It is a well-known fact that for the former (λμ+, λμ– ) and average (𝐼μ+ , 𝐼μ–) Lyapunov exponents of (𝒜ℤ, 𝐹), the inequalities 𝐼μ+ ≤ λμ+ , 𝐼μ– ≤ λμ– , ℎμ(𝒜ℤ, 𝐹) ≤ (𝐼μ+ , 𝐼μ–) · ℎμ(𝒜ℤ, σ), ℎ(𝒜ℤ, 𝐹) ≤ (λμ+ + λμ– ) · ℎμ(𝒜ℤ, σ) hold. Furthermore, under the above assumptions, there are examples of (𝒜ℤ, 𝐹) such that the average Lyapunov exponents provide a better upper bound for ℎμ(𝒜ℤ, 𝐹) than the former ones [P. Tisseur, Nonlinearity 13 (2000)]. In this paper we prove that under somewhat stronger assumptions, the average and former Lyapunov exponents can provide at least as excessive a real upper bound for ℎμ(𝒜ℤ, 𝐹) and ℎ(𝒜ℤ, 𝐹), respectively, as we choose it to be.
Keywords: cellular automata, Lyapunov exponents, entropy
Mathematics Subject Classification:
Primary 37B15, 37B10, 37A35; Secondary 37B40