= Solution
A map is <Devaney-chaotic> on $\Lambda$ when it is <topologically transitive>, it has <dense periodic points> in $\Lambda$, and it has <sensitive dependence on initial conditions>. A one-dimensional map is <Glendinning-chaotic> when some iterate has a <horseshoe for an interval map>: there are an interval $J$ and two disjoint subintervals $K_0,K_1\subseteq J$ such that $F^n(K_0)=F^n(K_1)=J$.
The map $F(x)=ax\pmod1$ is the <beta transformation> with slope $a>1$. On every interval of continuity of $F^n$, it is affine with slope $a^n$. The endpoints of these intervals are iterated preimages of the discontinuity set. Choosing $n$ with $a^n>2$, the standard full-branch lemma for the beta transformation supplies two disjoint continuity intervals $K_0,K_1$ on which $F^n$ maps bijectively onto $[0,1)$. Thus $F^n$ has a horseshoe, proving Glendinning chaos.
For Devaney chaos one may either use the standard theorem that a full-branch expanding interval map is Devaney-chaotic, or verify its ingredients directly. Every nonempty open interval expands by the factor $a$ on each uninterrupted iterate; after subdivision at preimages of discontinuities, one of its images contains a full branch and hence eventually covers $[0,1)$. This proves topological transitivity. Applying an inverse branch of a sufficiently high iterate to a full branch gives a contraction whose unique fixed point is periodic; choosing the branch inside any prescribed open interval proves that periodic points are dense. Finally, two sufficiently close points that follow the same branch separate by the factor $a$ on each iterate until their distance exceeds a fixed positive constant, proving sensitive dependence. Hence $F$ is also Devaney-chaotic for every real $a>1$.
Back to article page
