Asymptotic separation of periodic orbits and fractal dimension

Abstract

For a totally bounded metric dynamical system $(X, d, T)$ we introduce a new critical value $\mathfrak s(X, d, T)$ which quantifies the asymptotic separation of periodic orbits. More precisely, $\mathfrak s(X,d,T)$ is defined to be the supremum of all $s\ge 0$ for which there exists a sequence of periodic orbits $\{\mathcal O_k\}_{k=1}^\infty$ in $(X,d,T)$ such that \[ \lim_{k\to \infty} \#\mathcal O_k = +\infty \quad \text{and} \quad \liminf_{k\to\infty}\#\mathcal O_k\cdotη(\mathcal O_k)^s >0, \] where $η(\mathcal O_k)$ denotes the smallest distance between distinct points in $\mathcal O_k$. When the dynamical system is induced by a self-similar iterated function system $\mathcal F$ satisfying the strong separation condition, we prove that this critical value is equal to the Hausdorff dimension $s_{\mathcal F}$ of its self-similar attractor. Furthermore, under a quantitative separation condition on periodic orbits we show that the associated empirical periodic measures converge weakly to the normalized $s_{\mathcal F}$-dimensional Hausdorff measure on the self-similar attractor.