Topological expansive Lorenz maps with a hole at critical point

Abstract

Let $f$ be an expansive Lorenz map and $c$ be the critical point. The survivor set is denoted as $S_f(H):=\{x\in [0,1]:f^n(x)\notin H,\forall n\ge 0\}$, where $H$ is a open subinterval. Here we study the hole $H=(a,b)$ with $a\le c\le b$ and $a\ne b$. We show that the case $a=c$ is equivalent to the hole at $0$, the case $b=c$ equals to the hole at $1$. We also obtain that, given an expansive Lorenz map $f$ with a hole $H=(a,b)$ and $S_f(H)\{0,1\}$, then there exists a Lorenz map $g$ such that ${\tilde{S}}_f(H)\setminus \Omega (g)$ is countable, where $\Omega (g)$ is the Lorenz-shift of $g$ and ${\tilde{S}}_f(H)$ is the symbolic representation of $S_f(H)$. Let $a$ be fixed and $b$ varies in $(c,1)$, we also give a complete characterization of the maximal interval $I(b)$ such that for all $\epsilon \in I(b)$, $S_f(a,\epsilon )=S_f(a,b)$, and $I(b)$ may degenerate to a single point $b$. Moreover, when $f$ has an ergodic acim, we show that the topological entropy function ${\lambda }_f(a):b\mapsto h_{top}(f|S_f(a,b))$ is a devil staircase with $a$ being fixed, so is ${\lambda }_f(b)$ if we fix $b$. At the special case $f$ being intermediate $\beta$-transformation, using the Ledrappier-Young formula, we obtain that the Hausdorff dimension function ${\eta }_f(a):b\mapsto {\dim }_{\mathcal{H}}(S_f(a,b))$ is a devil staircase when fixing $a$, so is ${\eta }_f(b)$ if $b$ is fixed. As a result, we extend the devil staircases in \cite{Urbanski1986,kalle2020,Langeveld2023} to expansive Lorenz maps with a hole at critical point.

18pages