Invariant measures for parabolic IFS with overlaps and random continued fractions

K. Simon, B. Solomyak, M. Urbański
2001 Transactions of the American Mathematical Society  
We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no "overlaps." We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant
more » ... the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold. 5145 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5146 K. SIMON, B. SOLOMYAK, AND M. URBAŃSKI √ α 2 + 4α)], so the question becomes more delicate. Let χ α be the top Lyapunov exponent of the random matrix product (1.1). Lyons [Ly] proved that ν α is singular for all α ∈ (α c , 0.5] where α c ∈ (0.2688, 0.2689) is the only positive number satisfying log 2 = 2 χ αc . Absolute continuity was not proved for any value of α, but Lyons conjectured that ν α is absolutely continuous for all α sufficiently close to zero. We make progress on this conjecture and show that the threshold α c is sharp. In fact, we prove that ν α is absolutely continuous for a.e. α ∈ (α 0 , α c ), for some α 0 . (The value of α 0 = 0.215, that we obtain, has no special significance; we still don't know if the result holds for α 0 = 0.) This problem can be recast in the framework of iterated function systems (IFS). The measure ν α is an invariant measure for the IFS Φ α = {φ α 1 , φ 2 } := { x+α x+α+1 , x x+1 }. The measure ν α is supported on the limit set of the IFS, defined as the unique non-empty compact set satisfying J α = φ α 1 (J α ) ∪ φ 2 (J α ). For α ∈ (0, 0.5] we have J α = X α , an interval, and for α ∈ (0, 0.5) the intersection φ α 1 (J α ) ∩ φ 2 (J α ) is itself a non-empty interval (see Figure 1 for the case α = 1 4 ). Thus, we say that this IFS has an overlap. Another complication is that this IFS is parabolic (therefore, not strictly contracting), because φ 2 has a neutral fixed point at x = 0. Our approach is to consider Φ α as a family of IFS depending on a parameter and establish results for a.e. parameter value. Earlier work in this framework revealed the importance of a transversality condition in the parameter dependence, see [PoS, So1, PSo1, PSo2, So2, SSo, SSU1] . We consider more general parabolic IFS; they are defined precisely in the next section. Projecting an ergodic shift-invariant measure µ from the symbolic space to the limit set, we obtain an invariant measure ν = ν(Φ, µ) for the IFS. One can consider the entropy h µ and
doi:10.1090/s0002-9947-01-02873-2 fatcat:njb6pnj6cjhkdf7urqm3insbhq