Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps (Q2730707)

Let \(X\) be a compact subset of \(\mathbb R^d\) and \(\mathcal Z\) be a finite collection of bounded open polytopes \(Z_1,\dots ,Z_N\) such that \(X= \text{cl}(\bigcup_{i=1}^N Z_i)\). A map \(T:X\to X\) is piecewise affine if there are invertible affine maps \(T_i:\mathbb R^d \to \mathbb R^d\) such that \(T|Z_i=T_i|Z_i\) for \(i=1,\dots ,N\). Moreover, \(T\) is said to be generic if NEWLINE\[NEWLINE\lim_{n\to \infty} \tfrac{1}{n} \log \max_{x\in X} \#\{Z\in \mathcal Z_n\mid x \in \operatorname {cl} Z\}=0,NEWLINE\]NEWLINE where \(\mathcal Z_n=\{A_0\cap T^{-1}A_1 \cap \dots \cap T^{-n+1}A_{n-1}\neq \emptyset\mid A_0,\dots ,A_{n-1}\in \mathcal Z\}\). NEWLINENEWLINENEWLINEThe main result (Theorem 1) of the paper states that for a piecewise affine and eventually expanding map \(T\), if \(d=2\) or \(T\) is generic then (i) the Perron-Frobenius operator \(\mathcal L\) of \(T\) is a bounded linear operator on the space of bounded variations with spectral radius 1; (ii) the associated dynamical zeta function \(\zeta(z)\) is meromorphic in \(W=\{z\mid |z|<\theta^{-1}\}\) where the essential spectral radius \(r_{\text{ess}}(\mathcal L)\leq \theta <1\); (iii) \(z_0 \in W\) is a pole of \(\zeta\) of multiplicity \(p\) iff \(p=\dim (\bigcup_{n=1}^{\infty} \ker (Id - z_0\mathcal L)^n\). NEWLINENEWLINENEWLINEThis gives a partial generalization of the 1-dimensional result of \textit{V. Baladi} and \textit{G. Keller} [Commun. Math. Phys. 127, 459-478 (1990; Zbl 0703.58048)].