Stationary Measures for Projective Transformations: The Blackwell and Furstenberg Measures

Abstract

In this paper we study the Blackwell and Furstenberg measures, which play an important role in information theory and the study of Lyapunov exponents. For the Blackwell measure we determine parameter domains of singularity and give upper bounds for the Hausdorff dimension. For the Furstenberg measure, we establish absolute continuity for some parameter values. Our method is to analyze linear fractional iterated function schemes which are contracting on average, have no separation properties (that is, we do not assume that the open set condition holds, see Hutchinson in Indiana Univ. Math. J. 30:713–747, 1981) and, in the case of the Blackwell measure, have place dependent probabilities. In such a general setting, even an effective upper bound on the dimension of the measure is difficult to achieve.

Appendix

Here we briefly summarize the results of the papers [19] and [20] used in this note. We are given the family $\varPhi^{\mathbf{t}}= \{\phi_{1}^{\mathbf {t}},\dots,\phi_{m}^{\mathbf{t}} \}_{\mathbf{t}\in U}$ of hyperbolic IFS on a compact interval X⊂ℝ. (Hyperbolic means that there exist 0<c ₁<c ₂<1 such that for the $\mathcal{C}^{2}$ maps $\phi_{k}^{\mathbf{t}}$ we have $c_{1}<|(\phi_{k}^{\mathbf{t}})'(x)|<c_{2}$ for all k,t,x.) We assume that the parameter domain U⊂ℝ^d is a bounded open set with smooth boundary. As usual we denote the natural projection from Σ:={1,…,m}^ℕ to X by

$$\pi_\mathbf{t}(\mathbf{i}):= \lim _{n\to\infty} \phi_{i_1}^{\mathbf{t}}\circ\cdots\circ \phi_{i_n}^{\mathbf{t}}(x), $$

where x∈X is arbitrary. In this section we assume that we are given a σ-invariant ergodic measure μ on Σ and we always write

$$\nu_\mathbf{t}:=\pi_\mathbf{t}\mu $$

for its push down measure. The next theorem shows how the dimension and absolute continuity of the measure ν _t depends on the ratio of the entropy h(μ) of μ and the Lyapunov exponent λ _t of the measure ν _t defined as

$$\lambda_\mathbf{t}:=\int\phi^\mathbf{t}_{i_1}\bigl( \pi_\mathbf {t}(\sigma \mathbf{i})\bigr) \,d\mu(\mathbf{i}). $$

Below we often use the notation

$$\varSigma_2:= \bigl\{(\mathbf{i},\mathbf{j})\in\varSigma\times \varSigma:\ i_0\ne j_0 \bigr\}, \qquad f_{\mathbf{i},\mathbf{j}}(\mathbf{t}):= \pi_\mathbf{t}(\mathbf{i})- \pi_\mathbf{t}(\mathbf{j}). $$

We recall for the reader how the following theorem is proved in [19] and [20]:

Theorem 29

(Simon, Solomyak and Urbanski)

We assume that

(H):

$$\forall(\mathbf{i},\mathbf{j})\in\varSigma_2, \ \forall\mathbf{t}\in \overline{U}\colon \quad \bigl\|\nabla f_{\mathbf{i},\mathbf{j}} (\mathbf{t})\bigr\|>0.$$

Then

• (1) For Leb _d almost all t∈U, $\dim_{\mathrm{H}}\nu_{\mathbf{t}}=\min \{\frac{h(\mu)}{-\lambda_{\mathbf{t}}},1 \}$;

• (2) ν _t ≪Leb for Leb _d almost all t∈{t∈U:−h(μ)/λ _t >1}.

We start with a lemma from [19]. For a set F⊂ℝ^d let N _r (F) be the minimal number of balls needed to cover the set F.

Lemma 30

(See [19, Lemma 7.3].)

Let U⊂ℝ^d be as above. Suppose that f is a $\mathcal {C}^{1}$ real-valued function defined in a neighborhood of the closure of U such that for some i∈{1,…,d} there exists an η>0 satisfying

$$ \mathbf{t}\in U, \quad \bigl \vert f(\mathbf{t})\bigr \vert \leq\eta \quad \Longrightarrow \quad \frac{\partial f(\mathbf{t})}{\partial t_i}\geq\eta. $$

(30)

Then there exists C=C(η) such that

$$ N_r \bigl( \bigl\{ \mathbf{t}\in U:\ \bigl \vert f( \mathbf{t})\bigr \vert \leq r \bigr\} \bigr)\leq c\cdot r^{1-d}, \quad \forall r>0. $$

(31)

For every (i,j)∈Σ ₂ we may apply this lemma for f=f _i,j (t) since (H) implies that (30) holds. Write η _i,j and C _i,j for the corresponding constants. For compactness $\underline{\eta}:=\min_{(\mathbf{i},\mathbf{j})\in\varSigma_{2}} \eta_{\mathbf{i},\mathbf{j}}>0$ and $\overline{C}:=\max_{(\mathbf{i},\mathbf{j})\in\varSigma_{2}} C_{(\mathbf{i},\mathbf{j})}<\infty$. So,

$$\forall(\mathbf{i},\mathbf{j})\in\varSigma_2\colon\quad N_r \bigl( \bigl\{\mathbf{t}\in U: |f_{\mathbf{i},\mathbf{j}}|\leq r \bigr\} \bigr) \leq\overline{C} \cdot r^{1-d}. $$

The last statement is called the strong transversality condition (cf. [19, p. 454]) which clearly implies that the transversality condition holds: There exists a $\widetilde{c}>0$ such that

$$ \forall r>0\colon \quad \mathit{Leb}_d \bigl\{\mathbf{t}\in U: \vert f_{\mathbf{i},\mathbf{j}}\vert \leq r \bigr\}\leq\widetilde{c}\cdot r. $$

(32)

Then we can apply [20, Theorem 7.2] which immediately yields the assertion of our theorem.