Arnold's conjecture on exponential growth of periodic points in typical families

Let $f$ be a $C^\infty$ mapping, and let $P_f(n)$ denote the number of isolated points of period $n$. In a typical sufficiently parameterized family $(f_a)$, interpret “almost all” parameter values with respect to Lebesgue measure. Arnold's periodic-point growth conjecture. For almost every parameter value in each typical family, there is an exponential function of the period bounding the number of periodic points: $P_{f_a}(n)$ grows not faster than some exponential function of $n$. The source attributes this question to Smale and Arnold. It notes that the assertion is known in dimension one and in some real-analytic settings, whereas finite-smoothness generic results in higher dimensions give super-exponential growth in other senses; the stated typical-family conjecture remains unresolved in the supplied text.