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A map on a metric space is chaotic in Devaney's sense when it is topologically transitive, its periodic points are dense, and it has sensitive dependence on initial conditions.

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 $F(x)=ax(\mathrm{mod}1)$ 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 $a>1$.

A map is Devaney chaos when:

Write a point as a binary expansionApart from the harmless choice of expansion at dyadic rationals, the binary shift representation of the doubling map is

Every open interval contains a binary cylinder specified by a finite initial word. Given cylinders $U$ and $V$, choose a binary sequence beginning with the word for $U$ and place the word for $V$ after it. A suitable iterate shifts the second word to the front, proving topological transitivity.

Given any cylinder, repeat its defining word forever. The resulting point is periodic and lies in that cylinder, so periodic points are dense.

Finally, given $x$ and any neighbourhood, choose $N$ so large that changing only digits after the first $N$ stays inside that neighbourhood. Choose the later tail so that after $N$ shifts it is either $0$ or $3/4$, whichever is farther from $F^N(x)$. The separation is at least $3/8$, so any smaller fixed constant, for example $\delta =1/4$, proves sensitivity. Hence the doubling map is chaotic in Devaney's sense.