Symbolic Dynamics and Markov Structure for Tent-Type Maps

Abstract

We study a parameter-dependent family of tent-type maps and focus on the distinguished parameter value $\omega_0=\frac{1+\sqrt5}{2}$. After normalizing the dynamics to the unit interval, we show that the backward orbit of the unique turning point generates a Markov partition with a Fibonacci-type combinatorial structure. This yields an explicit symbolic model: the normalized map is (up to a standard countable exceptional set) topologically conjugate to the golden mean shift. We also determine an invariant probability measure that is absolutely continuous with respect to Lebesgue measure by solving the transfer-operator fixed point equation for the corresponding density. Finally, using the Markov/symbolic description, we compute escape rates for Markov holes.