Aperiodic sequences and aperiodic geodesics (Q2928256)

The authors introduce a quantitative condition to describe the aperiodicity of an orbit in a dynamical system. With respect to this condition, it is shown that there exist sequences in the Bernoulli shift and geodesics on closed hyperbolic manifolds being as aperiodic as possible. A brief description of the main results (in fact shown in greater generality) goes as follows:NEWLINENEWLINE(1) In discrete time, for an element \(w\) in the set \(\Sigma\) of all bi-infinite sequences in a finite set of \(k>1\) elements, define the recurrence time NEWLINE\[NEWLINE R_w^i(l):=\min\{s\geq 1:\,[w(i+s)\ldots w(i+s+l)]=[w(i)\ldots w(i+l)]\} NEWLINE\]NEWLINE and \(R_w(l):=\min_{i\in{\mathbb Z}}R_w^i(l)\), whose growth rate is understood as a measure of aperiodicity in \(w\). Given a non-decreasing function \(\varphi:{\mathbb N}_0\to[0,\infty)\) with NEWLINE\[NEWLINE \limsup_{l\to\infty}\frac{\ln\varphi(l)}{l}\leq\delta\ln k NEWLINE\]NEWLINE for some \(\delta\in(0,1)\), then there exists a non-negative integer \(l_0=l_0(\varphi,k,\delta)\) and a sequence \(w\in\Sigma\) such that \(R_w(l)\geq\varphi(l)\) for all \(l_0\leq l\).NEWLINENEWLINE(2) In continuous time, let \(M\) be a closed hyperbolic manifold having dimension \(n>1\), \(i_M>0\) denote the injectivity radius on \(M\) and suppose that \(d\) is the Riemannian distance function on \(M\). Given a unit speed geodesic \(\gamma:{\mathbb R}\to M\), the recurrence time \(R_\gamma^{t_0}:[0,\infty)\to[i_M/2,\infty]\) at time \(t_0\) reads as NEWLINE\[NEWLINE R_\gamma^{t_0}(l):=\inf\{s>i_M/2:\,d(\gamma(t_0+t),\gamma(t_0+s+t))<i_M/2\text{ for all }t\in[0,l]\} NEWLINE\]NEWLINE and \(R_\gamma(l):=\inf_{t_0\in{\mathbb R}}R_\gamma^{t_0}(l)\). If \(i_M>2\ln 2\) and \(\varphi:[0,\infty)\to[0,\infty)\) is a non-decreasing function satisfying NEWLINE\[NEWLINE \limsup_{l\to\infty}\frac{\ln\varphi(l)}{l}\leq\delta(n-1) NEWLINE\]NEWLINE for some \(\delta\in(0,1)\), then there exists a \(l_0=l_0(\varphi,\delta,n,i_M)\geq 0\) and a unit speed geodesic \(\gamma:{\mathbb R}\to M\) such that \(R_\gamma(l)\geq\varphi(l)\) for all \(l_0\leq l\).NEWLINENEWLINERather than working with the Bernoulli shift on \(\Sigma\), these considerations are generalized to describe \(F\)-aperiodicity of orbits to (semi-)dynamical systems on compact metric spaces, where \(F\) is a non-increasing real function (see the paper for details).