Let \(T:M'\to M'\) be a topologically transitive Anosov diffeomorphism of class \(C^{1+\varepsilon}\) on a compact Riemannian manifold \(M'\). Let \({\mathcal R} =(R_1, \dots, R_{I'}) \) be a Markov partition on \(M'\). Let \(I<I'\), \(H=\bigcup^{I'}_{i=I+1} (\text{int} R_i)\), \(M=M' \setminus H\).
The dynamics of \(T\) on \(M\), thinking of \(H\) as a ‘hole’ into which some points are mapped and disappear, is studied. It is assumed that the symbolic dynamics generated by the partition \(\{R_1, \dots, R_I\}\) is a topologically mixing subshift of finite type. Denote \(M_n= \bigcap^n_{i=0} T^iM\), \(M_{-n} =\bigcap^n_{i=0} T^{-i}M\), \(M_+ =\bigcap_{n\geq 1} M_n\), \(M_-= \bigcap_{n\geq 1} M_{-n}\), \(\Omega= M_+ \bigcap M_-\). \({\mathcal U}' =\bigvee^\infty_{n=0} T^n {\mathcal R}'\), \({\mathcal S}' =\bigvee^\infty_{n=0} T^{-n} {\mathcal R}'\) are the partitions of \(M'\) into unstable and stable fibers, respectively. The restrictions of \({\mathcal U}'\) to \(M\) \((M_+)\) are denoted by \({\mathcal U}\) \(({\mathcal U}_+)\), respectively. A conditionally invariant measure is defined.
Some of the main results of the paper are the following. There is a unique conditionally invariant family of probability measures \(\nu^u_U\), on fibers \(U\in {\mathcal U}_+\). Denote by \({\mathcal M}^n_+\) the class of Borel measures supported on fibers \(U\in {\mathcal U}_+\) coinciding with \(\nu^n_U\). The map \(T\) has a unique conditionally invariant measure \(\mu_+ \in {\mathcal M}^n_+\). The sequence of measures \(T_*^{-n} \mu_+\) \((T_* \mu= \mu T^{-1})\) weakly converges to a probability measure \(\eta+\) supported on \(\Omega\). The measure \(\nu_+\) is invariant, ergodic, mixing, \(K\)- mixing and Bernoulli. Its correlations decay exponentially fast and it satisfies the central limit theorems. \(\eta_+\) is Gibbs’ measure. The variational principle for \(\eta_+\) holds.