Lasota-Yorke maps with holes: Conditionally invariant probability measures and invariant probability measures on the survivor set
Let \(X_0=I\backslash Y\), and \(X_n=\cap_{i=0}^nT^{-i}X_0\). Then \(X_\infty\) is the survivor. Condition 0. The potential satisfies \(\inf g^0>0\), \(g^0\in BV\), contracting, and there exists a \(g^0\)-conformal probability measure \(m\). Let \({\mathcal L}_0\) be the usual Perron-Frobenius operator associated with \(T\), and \({\mathcal L}(f)={\mathcal L}_0(f1_{X_0})\). Condition 1. Let \(D_n=\{x\in I:{\mathcal L}^n1(x)\neq 0\}\). Assume \(D_\infty=\cap_nD_n\neq\emptyset\). Let \(\Lambda(f)=\lim_{n\to\infty}\inf_{x\in D_n} {{\mathcal L}^nf(x)\over{\mathcal L}^n1(x)}\). The main tool to prove the theorems is the Lasota-Yorke type inequality on the cone \(C_a=\{h\in BV: h\not\equiv 0\), \(h\geq 0\), \(\vee h\leq a\Lambda(h)\}\): \[ \bigvee{\mathcal L}^nh\leq C_\theta\theta^n\bigvee h+K_n\Lambda(|h|). \] To estimate ``bad'' subintervals for which \(\Lambda(1_Z)=0\), another assumption (Condition 2) is needed. We omit the details. Theorem A. Assume conditions 0, 1 and 2 are satisfied. Then there exists a unique conditionally invariant probability measure \(\nu=hm\). There exists a unique probability measure \(\mu\) supported in \(X_\infty\), and the measure \(\nu(A|X_{n-1})\) converges to \(\lambda(A)=\int_Ah d\mu\) in exponential order. The measure \(\mu\) is determined by the Perron-Frobenius operator such as \(\mu({\mathcal L}f)=\rho\mu(f)\) \((f\in BV)\) with some constant \(\rho\). Definition. Let \(Z^{(n)}\) be a set of subintervals associated with words with length \(n\). \(T\) has large images if \(\inf_n\inf_{Z\in Z^{(n)}}m(T^nZ)>0\), and \(T\) has large images with respect to \(Y\) if for all \(n\), for all \(Z\in Z^{(n)}\), \(Z\cap X_\infty\neq\emptyset\), \(T^n(Z\cap X_{n-1})\supset X_\infty\). When one considers a potential \((g^0)^t\) \((0\leq t\leq 1)\) instead of \(g^0\), next Theorem B follows. Theorem B. Let \(g^0=|T'|^{-1}\). Assume that for all \(0\leq t\leq 1\), conditions 0, 1 and 2 are satisfied. Then there exists a unique \(0<t_0\leq 1\) such that \(\rho_t>1\) for \(0\leq t<t_0\) and \(\rho_t>1\) for \(1\geq t>t_0\). If \(T\) has large images and has large images with respect to \(Y\), then \(HD(X_\infty)=t_0\). When a hole is small, one can apply a perturbative approach, and get: Theorem C. Assume \(g^0\) satisfies condition 0. If \(T\) has a unique invariant measure \(\mu_0\) absolutely continuous with respect to \(m\), and \((I,T,\mu_0)\) is mixing, then there exists \(\varepsilon>0\) such that for each hole \(Y\) \(m(Y)<\varepsilon\), the conclusions of theorem A are valid.
conditionally invariant probability measure
scientific article; zbMATH DE number 1915675