Dynamically defined recurrence dimension (Q1347789)

Two new topological invariants, so-called polynomial recurrence dimension and polynomial (topological) entropy, are introduced and discussed in this paper. The first one is a variant of the Hausdorff dimension, with the diameter of a set replaced by a gauge function (\(1/t\)) of the smallest return time of a set into itself. More precisely, let \((X,f)\) be a zero-dimensional dynamical system (\(X\) is homeomorphic to a closed subset of \(A^{\mathbb N}\) with \(A\) finite and \(f:X\rightarrow X\) is continuous), let \({\mathcal V}=\{V_a:a\in A\}\) be a partition of \(X\) by clopen sets, and let \(Y\subset X\), \(\alpha>0\). Write \(\text{diam}({\mathcal V})=\max\{\text{diam}(V):V\in {\mathcal V}\}\), \({\mathcal V}^n=\{\bigcap_{i=0}^{n-1} f^{-i}(V_{a_i}): a_1a_2\cdots a_n\in A^n\}\) and \(\tau(Y)=\min\{k>0:f^k(Y)\cap Y\neq \emptyset\}\). Now, let \[ \underline{M}_p(Y,f,{\mathcal V},\alpha)=\liminf_{n\rightarrow \infty} \sum_{V\in {\mathcal V}^n, V\cap Y\neq \emptyset} \tau(V)^{-\alpha}, \] \[ \overline{M}_p(Y,f,{\mathcal V},\alpha)=\limsup_{n\rightarrow \infty} \sum_{V\in {\mathcal V}^n, V\cap Y\neq \emptyset} \tau(V)^{-\alpha}. \] Next we write \(\underline{m}_p(Y,f,\alpha)=\limsup_{\varepsilon\rightarrow 0} \{\underline{M}_p(Y,f,{\mathcal V},\alpha):\text{diam}({\mathcal V})<\varepsilon\}\) and define the lower polynomial recurrence dimension of \(Y\) by \(\underline{r}_p(Y,f)=\sup\{\alpha>0:\underline{m}_p(Y,f,\alpha)=\infty\}\). If in the notation above we replace \(\underline{M}_p\) and \(\underline{m}_p\) by \(\overline{M}_p\) and \(\overline{m}_p\) then we get the upper polynomial recurrence dimension \(\overline{r}_p(Y,f)\) of \(Y\). Lower and upper polynomial entropies \(\underline{h}_p(X,f)\) and \(\overline{h}_p(X,f)\) are defined for any dynamical system \((X,f)\) (it is only required that \(X\) is compact metric) and apply when usual topological entropy is zero but there is still a polynomial growth in the cardinality of the coverings by open sets which are used in the standard definition. We refer to the paper for the precise definition which, in the case when \((X,f)=\Sigma\) is a subshift of finite type, amounts to \(\underline{h}_p(\Sigma)=\liminf_{n\rightarrow \infty} \frac{\log P(n)}{\log n}\) and \(\overline{h}_p(\Sigma)=\liminf_{n\rightarrow \infty} \frac{\log P(n)}{\log n}\) for an appropriately defined ``complexity function'' \(P(n)\) of the subshift. It is worth emphasizing that usual entropy is then defined by \(h(\Sigma)=\lim_{n\rightarrow \infty} \frac{\log P(n)}{n}\). Properties of these notions and the relationship between them are investigated (in what follows \(r_p\) stands for both \(\underline{r}_p\) and \(\overline{r}_p\), and similarly for \(m_p\)). In particular: 1. If \((X,f)\) is minimal then \(r_p(Y,f)=\inf\{\alpha>0: m_p(Y,f,\alpha)=0\}\). 2. The set function \(m_p(\cdot,f,\alpha)\) is a Borel measure. 3. If \(Y\) is a closed invariant subset for \((X,f)\) then \(m_p(Y,f|_Y,\alpha)\leq m_p(Y,f,\alpha)\leq m_p(X,f,\alpha)\) and \(r_p(Y,f|_Y)\leq r_p(Y,f)\leq r_p(X,f)\). 4. If \(\text{Per}(n)=\text{card} \{x\in X:f^n(x)=x\}\) then \(\overline{r}_p(X,f)\geq \limsup_{n\rightarrow \infty} \frac{\log \text{Per}(n)}{\log n}\) and \(\underline{r}_p(X,f)\geq \liminf_{n\rightarrow \infty} \frac{\log \text{Per}(n)}{\log n}\). 5. If \(\Omega\) is the set of nonwandering points of \((X,f)\) then \(r_p(X,f)=r_p(\Omega,f)\). 6. If \((X,f)\) is minimal then \(\underline{r}_p(X,f)\geq 1\). 7. For every subshift \(\Sigma\) we have \(\overline{r}_p(\Sigma)\leq \overline{h}_p(\Sigma)+1\). Finally, recurrence dimensions of Sturmian subshifts are computed and some examples concerning Toeplitz subshifts are given.