Spectral and dynamical results related to certain non-integer base expansions on the unit interval

We consider certain non-integer base β -expansions of Parry’s type and we study various properties of the transfer (Perron–Frobenius) operator P Lᵖ([0,1])→ Lᵖ([0,1]) with p≥ 1 and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these β -expansions.We show that if f is Lipschitz, then the iterated sequence \{PN f\}N≥ 1 converges exponentially fast (in the L¹ norm) to an invariant state corresponding to the eigenvalue 1 of P . This “attracting” eigenvalue is not isolated: for 1≤ p≤ 2 we show that the point spectrum of P also contains the whole open complex unit disk and we explicitly construct an eigenfunction for every z with |z|<1 .