We have calculated photoionization cross section of endohedral atoms A@C_N. For the first time, we took into account simultaneously the polarizability of the fullerene electron shell C_N that modifies the incoming photon beam and the modification of the one-electron wave functions of the caged atom A under the action of C_N and atom A polarization potentials. The properly modified one-electron wave functions became a starting point of the account of the multi-electron correlations in the frame of the random phase approximation with exchange (RPAE). As concrete objects, we have considered Ar and Xe atoms inside fullerene C₆₀. We demonstrate that the effect of polarizability of the fullerenes C_N electron shell strongly affects the caged atom A photoionization cross section due to dynamic C_N polarization, i.e. via the modification of the incoming photon beam that acts upon atom A, located inside C_N, as a resonator or amplifier. To study this effect qualitatively, we assume the validity of a strong inequality r_A<R and U_F = -α_c/2(r²+R²)² for r>R. Along with С_N polarization, we took into account in a similar approximation the polarization potential of the atom A itself. In the calculations performed we described the fullerenes shell by a Lorentz-shape potential that represents the charge density distribution in CN qualitatively better than previously used by most researchers square well potential. The results obtained demonstrate profound influence of static and dynamic polarizabilities at low photon energies. Obviously, the influence of the fullerenes shell dies out, as it should be, with photon energy growth.