36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

"Expanding Lorenz maps with slope greater than $\sqrt{2}$ are leo"

Bartłomiejczyk, Piotr

We prove that expanding Lorenz maps with slope greater than $\sqrt{2}$ are locally eventually onto (leo). To be more precise, recall that an expanding Lorenz map is a map $f\colon[0,1]\to[0,1]$ satisfying the following three conditions: $$ $$ 1.) there is a critical point $c\in(0, 1)$ such that $f$ is continuous and strictly increasing on $[0, c)$ and $(c, 1]$, $$ $$ 2.) $\lim_{x\to c^-}f(x)=1$ and $\lim_{x\to c^+}f(x)=f(c)=0$, $$ $$ 3.) $f$ is differentiable for all points not belonging to a finite set $F\subset[0, 1]$ and there is $\lambda>1$ such that $\inf{\{f'(x)\mid x\in[0,1]\setminus F\}}\ge\lambda$. $$ $$ Assume that $f$ is an expanding Lorenz map and $\beta=\inf{\{f'(x)\mid x\in[0,1]\setminus F\}}$. Let $f_0(x)=\sqrt{2}x+\frac{2-\sqrt{2}}{2} \pmod 1$. If $\beta\ge\sqrt{2}$ and $f\neq f_0$ then for every nonempty open subinterval $J\subset(0,1)$ there exists $n\in\mathbb{N}$ such that $f^n(J)\supset[0,1)$. This is joint work with Piotr Nowak-Przygodzki.

« back