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The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the finiteness property if there exists a periodic product which achieves this maximal rate of growth. J. C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real d × d matrices satisfies the finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the finiteness conjecture exist, showing in particular that there exists a family of pairs of 2 × 2 matrices which contains a counterexample. Similar results were subsequently given by V. D. Blondel, J. Theys and A. A. Vladimirov and by V. S. Kozyakin, but no explicit counterexample to the finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the set

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Abstract. We define a two-sided analog of the Erd6s measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erd6s measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erd6s and Lebesgue measures on T 2 that is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erd6s measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.

Let q ∈ (1, 2); it is known that each x ∈ [0, 1/(q − 1)] has an expansion of the form x = ∞ n=1 a n q −n with a n ∈ {0, 1}. It was shown in [P. Erdős, I. Joó, V. Komornik, Characterization of the unique expansions 1 = ∞ i=1 q −n i and related problems, Bull. Soc.Math. France 118 (1990) 377-390] that if q < ( √ 5 + 1)/2, then each x ∈ (0, 1/(q − 1)) has a continuum of such expansions; however, if q > ( √ 5 + 1)/2, then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535-543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m > 1 of expansions in base q. In particular, we show that if q < q 2 = 1.71 . . . , then each x has either 1 or infinitely many expansions, i.e., there are no such q in (( √ 5 + 1)/2, q 2 ). On the other hand, for each m > 1 there exists γ m > 0 such that for any q ∈ (2 − γ m , 2), there exists x which has exactly m expansions in base q.