Extensions of exact and K-mixing dynamical systems

Abstract

We consider extensions of non-singular maps which are exact, respectively K-mixing, or at least have a decomposition into positive-measure exact, respectively K-mixing, components. The fibers of the extension spaces have countable (finite or infinite) cardinality and the action on them is assumed surjective or bijective. We call these systems, respectively, fiber-surjective and fiber-bijective extensions. Technically, they are skew products, though the point of view we take here is not the one generally associated with skew products. Our main results are an Exact and a K-mixing Decomposition Theorem. The latter can be used to show that a large number of periodic Lorentz gases (the term denoting here general group extensions of Sinai billiards, including Lorentz tubes and slabs, in any dimension) are K-mixing.

Figures (6)

• Figure 1: A (proper) periodic Lorentz gas in $\mathbb{R}^2$.
• Figure 2: A Lorentz tube in the space $E^2 = \mathbb{R} \times \mathcal{T}^1$.
• Figure 3: A Lorentz slab in the space $E^3 = \mathbb{R}^2 \times \mathcal{T}^1$.
• Figure 4: Illustration of an argument for the proof of Proposition \ref{['prop-lg']}.
• Figure 5: A Lorentz tube with hard walls (cf. Remark \ref{['rk-2nd-last']}).

Theorems & Definitions (11)

• Lemma 2.1
• Proposition 2.2
• Theorem 2.3
• Corollary 2.4
• Theorem 2.5
• Corollary 2.6
• Lemma 3.1
• Proposition 4.1
• Remark 4.2
• Remark 4.3