# Random linear map contracting distances on the projective line

Let $$P^1$$ be the real projective line, identified with lines of $$\mathbb{R}^2$$. Let $$\mu$$ be a probability measure on invertible linear map of $$\mathbb{R}^2$$ and let $$(A_i)$$ be i.i.d. random variables distributed according to $$\mu$$. Let $$p,q\in P^{1}$$ and $$p_i=A_iA_{i-1}...A_1p$$, $$q_i=A_iA_{i-1}...A_1q$$. I feel that under quite mild assumption on $$\mu$$, we can have a that $$p_i$$ and $$q_i$$ get closer and closer to each other (with respect to the natural distances on $$P^1$$). But I cannot formulate or find a precise statement. Could someone propose one? Or give me a hint in what direction I should search?

Note that some conditions are needed: If the matrices $$A_i$$ are all rotation matrices then no contraction takes place. A relatively easy case is when all the matrices are strictly positive elementwise (it suffices that they be nonnegative with some product of them strictly positive.) In this case the matrices contract the Hilbert metric on projective space. See [1] for the relevant definitions and [2] for the application to random matrices. The preceding extends to the case when all matrices involved preserve a nontrivial cone, and some product of them maps this cone into its interior. In the general case, this is a basic question in the theory of random matrix products first developed in [3]. The contraction is guaranteed when the top lyapunov exponent is positive and simple, for a.e. starting vector. General conditions for this were given by Goldsheid and Margulis [4,5] following earlier work by Guivarch-Raugi. A nice survey is [6], see Sections 3 and 4 there that discuss the contraction.

[1] Seneta, E., 2006. Non-negative matrices and Markov chains. Springer Science & Business Media.

[2] Peres, Y., 1991. Analytic dependence of Lyapunov exponents on transition probabilities. In Lyapunov exponents (pp. 64-80). Springer, Berlin, Heidelberg.

[3] Furstenberg, Harry. "Noncommuting random products." Transactions of the American Mathematical Society 108, no. 3 (1963): 377-428.

[4] Goldsheid, I. Ya, and G. A. Margulis. "Lyapunov exponents of random matrices product." Usp. Mat. Nauk 44 (1989): 13-60.

[5] Goldsheid, I. Ya, and G. A. Margulis. "Conditions of simplicity of the spectrum of Lyapunov indices." In Dokl. Akad. Nauk SSSR, vol. 293, pp. 297-301. 1987.

[6] Guivarc'h, Yves. "On contraction properties for products of Markov driven random matrices." Журнал математической физики, анализа, геометрии 4.4 (2008): 457-489. http://www.mathnet.ru/php/getFT.phtml?jrnid=jmag&paperid=109&what=fullt http://www.mathnet.ru/links/f785a28766453ce0a3ab973677e3961c/jmag109.pdf