## Ergodic properties of Anosov maps with rectangular holes.(English)Zbl 0893.58035

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.

### MSC:

 37A99 Ergodic theory 37E99 Low-dimensional dynamical systems 37D99 Dynamical systems with hyperbolic behavior
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### References:

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