Abstract
Given any compact connected manifold \(M\), we describe \(C^2\)-open sets of iterated functions systems (IFS’s) admitting fully-supported ergodic measures whose Lyapunov exponents along \(M\) are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe \(C^1\)-open sets of IFS’s admitting ergodic measures of positive entropy whose Lyapunov exponents along \(M\) are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS’s on the flag manifold.
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Notes
A more sophisticated relation of this kind was obtained by Johnson et al. [14].
Lemma 10.2 below gives a practical criterion for positive minimality of an IFS.
All the measures we consider will be defined over the corresponding Borel \(\sigma \)-algebra.
The continuous-time version of Proposition 3.7 is sometimes called the Liao spectrum theorem; see e.g. [9].
This metric is obviously right-invariant. Actually, it is also left-invariant. Indeed, a calculation shows that \(\langle X, Y \rangle = - {{\mathrm{tr}}}XY/2\) for \(X, Y\in \mathfrak so (d)\), which is invariant under the adjoint action of the group, and therefore can be uniquely extended to a bi-invariant Riemannian metric. Another remark: this inner product is the Killing form divided by \(-2(d-2)\) (if \(d>2\)).
Those familiar with the QR algorithm will recognize this equation; see Remark 8.3.
Similar results are obtained in [22]; see Lemma 4.
We thank Carlos Tomei for telling us about the QR algorithm.
This field of horizontal subspaces is actually an Ehresmann connection on the principal bundle \({\check{\mathcal{F }}}M\).
As we mentioned in Remark 3.5, Oseledets [19] reduced the proof of his theorem to the triangular case.
Here we use that \(N\) is a connected Riemannian manifold (and not only a metric space).
Using this, it is easy to establish the existence of \(C^1\)-robustly positively minimal finitely generated IFS’s on any compact connected manifold. More interestingly, Homburg [13] shows that two generators suffice.
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We are grateful to the referee for some corrections.
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The authors received support from CNPq, FAPERJ, PRONEX (Brazil), Balzan–Palis Project, Brazil–France Cooperation Program in Mathematics, and ANR (France).
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Bochi, J., Bonatti, C. & Díaz, L.J. Robust vanishing of all Lyapunov exponents for iterated function systems. Math. Z. 276, 469–503 (2014). https://doi.org/10.1007/s00209-013-1209-y
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DOI: https://doi.org/10.1007/s00209-013-1209-y