Examples of Anosov Lie algebras

Abstract

We construct new families of examples of (real) Anosov Lie algebras starting with algebraic units. We also give examples of indecomposable Anosov Lie algebras (not a direct sum of proper Lie ideals) of dimension 13 and 16, and we conclude that for every n ≥ 6 with n = 7 there exists an indecomposable Anosov Lie algebra of dimension n.

Characterization of 9-dimensional Anosov Lie algebras

“…The properties of these type of algebraic units, studied in [17], can be used to give examples or to prove non-existence of Anosov Lie algebras. This approach was introduced in [11] and have also been used in [13], [14] and [18]. Using this, we prove that every 9-dimensional real Anosov Lie algebra, without an abelian factor, is either of type (6,3) or (3,3,3).…”

“…In this case, we may rearrange the basis so that λ 4 = λ −2 1 , λ 5 = λ −2 2 and λ 6 = λ −2 3 . Hence we can assume that the nonzero Lie brackets in n are given by (18) […”

“…Let n(a, b, c) denote the Lie algebra defined by (18). Due to our assumption of no abelian factor we have that abc = 0 and then n ≃ n(a, b, c) ≃ n(1, 1, 1) ≃ U 1 + m 10 (see Notation 3.6 and (m 10 )).…”

“…The first example of a non-toral nilmanifold admitting an Anosov automorphism was described by S. Smale ([20]). For many years only relatively few examples appeared in the literature, but in recent years families of nilmanifolds with Anosov automorphisms have been constructed showing that a complete classification does not seem to be possible (see [11,2,4,5,6,7,13,14,16,18,21]).…”

The geodesic flow on nilpotent Lie groups of steps two and three

“…Different authors studied dynamical aspects of geodesic flows on nil-and solv-manifolds (see [4,5]), manifolds which are locally homogeneous and obtained as compact quotients of nilpotent or solvable Lie groups. Other relations between geometrical and dynamical aspects on solvable or nilpotent Lie groups (or their quotients), were treated for instance in the references [6,7,8,9,10,17,18,20,21].…”