By Luis Barreira

ISBN-10: 3034802064

ISBN-13: 9783034802062

This self-contained monograph provides a unified exposition of the thermodynamic formalism and a few of its major extensions, with emphasis at the relation to measurement conception and multifractal research of dynamical platforms. particularly, the ebook considers 3 varied flavors of the thermodynamic formalism, particularly nonadditive, subadditive, and virtually additive, and offers a close dialogue of a few of the main major leads to the world, a few of them relatively contemporary. it is usually a dialogue of the main colossal purposes of those flavors of the thermodynamic formalism to measurement conception and multifractal research of dynamical systems.

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**Example text**

20) we successively obtain Rn (ϕ, ε) ≤ Zn (ϕ, U) ≤ enϕ(U) A ≤ enγ(U) Sn (ϕ, δ) ≤ e2nγ(U) B ≤e 2nγ(U) Rn (ϕ, δ). 21) 28 Chapter 2. 22) and similarly, lim lim sup ε→0 n→∞ 1 log Rn (ϕ, ε) ≤ PX (ϕ) n 1 log Sn (ϕ, δ) n 1 ≤ lim lim sup log Rn (ϕ, δ). 23). 7), and this completes the proof of the theorem. 3 is based on arguments in [5] and [39]. 2). 3 Variational principle We present in this section the variational principle for the topological pressure, which shows that the pressure of any continuous function ϕ is the supremum of the quantity hμ (f ) + X ϕ dμ over all f -invariant probability measures μ in X.

Clearly, the collection Γn covers X for each n ∈ N. 2. Equivalent deﬁnitions of pressure 25 for each U = (U1 , . . , Un ) ∈ Γn . 11) U ∈Γ we thus obtain n N (Γ ) ≤ exp −αm(Ui ) + ϕ(Ui ) = N (Γ)n . 12) i=1 Ui ∈Γ Given α > PX (ϕ, U), there exists m ∈ N and a collection Γ ⊂ covering X such that N (Γ) < 1. 12) that ∞ N (Γ∞ ) ≤ ∞ N (Γn ) ≤ n=1 N (Γ)n < ∞. n=1 Moreover, since Γ covers X, given p ∈ N and x ∈ X there exists U ∈ Γ∞ such that x ∈ X(U ) and p ≤ m(U ) < p + q. We denote by Γ∗p ⊂ Wp (U) the collection of all vectors U ∗ composed of the ﬁrst p elements of some of these U ∈ Γ∞ .

We ﬁrst recall some basic notions of symbolic dynamics, starting with the case of one-sided sequences. N For each κ ∈ N, we consider the set Σ+ of one-sided sequences κ = 1, . . , κ + of numbers in {1, . . , κ}. Given a sequence ω ∈ Σκ we write it in the form ω = i1 (ω)i2 (ω) · · · . L. 1007/978-3-0348-0206-2_3, © Springer Basel AG 2011 41 42 Chapter 3. The Case of Symbolic Dynamics + Moreover, we deﬁne the shift map σ : Σ+ κ → Σκ by σ(i1 i2 · · · ) = (i2 i3 · · · ). Now we introduce a distance, and thus also a topology in Σ+ κ .

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