We will denote the compact interval $[0,1]$ by $I$ and the unit square $[0,1]\times[0,1]$ by $I^2$. Triangular map on $I^2$ is a continuous map $F:I^2\to I^2$ of the form $F(x,y)=(f(x),g(x,y))$ where $f:I\to I$ and $g:I\times I\to I$ are continuous maps. Here $f:I\to I$ is called the base map and $g_x$ are called the fibre maps. $g_x:I\to I$ defined by $g_x(y)=g(x,y)$. All the maps of unit square having some kind of chaos that I have encountered in literature are triangular maps. The relevant definitions will be given at the end. Here is a list of publications to support my comment.
- B. Shanfelder, A. Crannell. Chaotic results for a triangular map of the square, Math. Mag., Issue 1, (2000), 13-20
(The paper above has a concrete construction of a devaney chaotic triangular map that is also mixing.)
- L. Alseda, S. F. Kolyada and J. Llibre, L. Snoha. Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., vol. 351, Issue 4, (1999), 1551–1573.
( The paper above contains a construction of a transitive triangular map, cf. Th1.7)
- F. Balibrea, L. Snoha. Topological entropy of Devaney chaotic maps, Topology and its Applications, Vol. 133, Issue 3, (2003), 225-239.
(This one has an abstract construction that any base map can be supplemented in the 2nd coordinate to create a devaney chaotic triangular map of the product space using category method)
- Z. Kocan. The problem of classification of triangular maps with zero topological entropy, Annales Mathematicae Silesianae, Issue 13, (1999), 181–192
(lemma 3.5 in above paper has a construction of a triangular map of the square with positive entropy but no homoclinic trajectory)
Someone can probably go on with the list but It seems that all of the 'chaotic' examples are triangular maps. So my question is the following:
Is there a continuous map $F:I^2\to I^2$ with 'chaotic properties' that is 'genuinely NOT a triangular map'?
CLARIFICATIONS:
By chaotic properties I mean any of the following: transitive, mixing, devaney chaotic on an infinite subset, having a horseshoe, having homoclinic trajectory or positive topological entropy.
By 'genuinely NOT a triangular map' what I mean is that there is not a compact set with non empty interior where $F$ agrees with some triangular map defined on the unit square and the dynamics is 'chaotic' on that set and outside it is extended to be continuous map from the unit square to itself.
I had another kind of related question to ask:
What happens to the dynamics if $F(x,y)=(f(x),g(x,y))$ is replaced by $F(x,y)=(g(x,y), f(x))$?
RELEVANT DEFINITIONS:
Let $(X,d)$ be a metric space in all of the following.
TOPOLOGICAL TRANSITIVITY: A function $f:X\to X$ is said to be topologically transitive if for any pair of non empty open sets $U$ and $V$ we can find a positive integer $n$ such that $f^n(U)\cap V\ne \emptyset$.
MIXING: We say that the map $f:X\to X$ is mixing if for any pair of non empty open sets $U$ and $V$ we can find a positive integer $N$ such that $f^n(U)\cap V\ne \emptyset$ for every $n\ge N$.
DEVANEY CHAOS: A continuous map $f:X\to X$ is said to be chaotic in the sense of Devaney if $f$ is topologically transitive and the periodic points of $f$ are dense in $X$.
HORSESHOE: We say that a map $f:X\to X$ has $n$-horseshoe, if there exists non-empty interior having compact subsets $J_1,\ldots,J_s$ with pairwise disjoint interiors such that $ \bigcap_{i=1}^s f(J_i) \supset \bigcup_{i=1}^s J_i$.
TOPOLOGICAL ENTROPY: $f: X \to X$ be a continuous map. Then the topological entropy of $f$ is denoted as $h_{\rm top}(f)$ and is defined by the following quantity, $h_{\rm top}(f)=\lim_{\epsilon \to 0+} \limsup\limits_{n \to \infty}{\frac{\log S_{n}(f, \epsilon)}{n}}.$ (One may refer to the book of Brin and Stuck)
