In 2014, Jeandel proved that two dynamical properties regarding Turing machines can be computable with any desired error $ϵ>0$, the Turing machine Maximum Speed and Topological Entropy. Both problems were proved in parallel, using equivalent properties. Those results were unexpected, as most (if not all) dynamical properties are undecidable. Nevertheless, Topological Entropy positiveness for reversible and complete Turing machines was shortly proved to be undecidable, with a reduction of the halting problem with empty counters in 2-reversible Counter machines. Unfortunately, the same proof could not be used to prove undecidability of Speed Positiveness. In this research, we prove the undecidability of Homogeneous Tape Reachability Problem for aperiodic and reversible Turing machines, in order to use it to prove the undecidability of the Speed Positiveness Problem for complete and reversible Turing machines.

在2014年，让德尔（Jeandel）证明了关于图灵机的两个动力学特性可以计算出任何期望的误差 $ϵ>0$图灵机的最大速度和拓扑熵。使用等效属性并行证明了这两个问题。这些结果是出乎意料的，因为大多数（如果不是全部）动力学特性是不确定的。尽管如此，可逆的和完整的图灵机的拓扑熵正数很快被证明是不可确定的，从而减少了2可逆计数器机器中空计数器的停止问题。不幸的是，不能使用相同的证据来证明速度肯定性的不确定性。在本研究中，我们证明了非周期性和可逆图灵机的均质胶带可达性问题的不确定性，以便用它来证明完整和可逆图灵机的速度正性问题的不确定性。