Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups

Abstract

Abstract. Let Fn be the free group of rank n, and ∂Fn its boundary (or space of ends).For any α ∈ Aut Fn, the homeomorphism ∂α induced by α on ∂Fn has at least two periodic points of period ≤ 2n. Periods of periodic points of ∂α are bounded above by a number Mn depending only on n, with log Mn ∼ n log n as n → +∞. Using the canonical Hölder structure on ∂Fn, we associate an algebraic number λ ≥ 1 to any attracting fixed point X of ∂α; if λ > 1, then for any Y close to X the sequence ∂α p (Y ) approaches X at about the same speed as e −λ p . This leads to a set of Hölder exponents Λ h (Φ) ⊂ (1, +∞) associated to any Φ ∈ Out Fn. This set coincides with the set of nontrivial exponential growth rates of conjugacy classes of Fn under iteration of Φ. Mathematics Subject Classification (2000). 20E05, 20F65.

Counting Growth Types of Automorphisms of Free Groups

“…We also explain how to determine the growth types (λ, m) with λ > 1 from the set of attracting laminations of α and the Perron-Frobenius eigenvalues. As shown in [12], the numbers λ may be viewed as Hölder exponents associated to periodic 3 points on the boundary of F n . Given α, the number e ′ of growth types (λ, m) with λ > 1 is bounded by e, and Theorems 1 and 2 are valid with e replaced by e ′ .…”

Infinite periodic points of endomorphisms over special confluent rewriting systems

“…Hence α ∈ Per(Φ) is an attractor for Φ if and only if it is a repeller for Φ −1 and vice-versa. The dynamical study of automorphisms of the free group has been carried on by different authors (e.g., [4,5,10,12,13,14]). In particular, it is known that: We intend to deal with a more general situation, going beyond the free group and beyond automorphisms.…”

“…The dynamical study of the automorphisms of a free group and their space of ends is a well established subject in discrete Dynamical Systems [4,5,10,12,13,14]. This paper constitutes an effort to study these problems in a more general setting, by considering monoids defined by certain types of rewriting systems instead of just free groups, and endomorphisms instead of automorphisms.…”

Free group automorphisms with parabolic boundary orbits

“…Levitt and Lustig have proved in [14] that there exists an integer p , which depends only on the rank N of F N , such that for all ' 2 Aut.F N /, the periodic points of x ' p are fixed points:…”

“…The dynamics of the map @' on @F N has been studied a lot; see Levitt and Lustig [13;14;15;16] and the author's thesis [10]. We give a survey of the known results relevant in our context in Section 3.…”