All Questions
Tagged with topological-dynamics chaos-theory
5 questions
2
votes
1
answer
51
views
Existence Of An Orbit of the Tent Map whose Closure is the Cantor Set [closed]
How to prove that
$$
T(x) = \frac{3}{2} - 3|x-\frac{1}{2}|,
$$
has at least one point with dense orbit with respect to its invariant middle third Cantor set?
Are there any basic solutions?
Thank you ...
- 410
3
votes
0
answers
104
views
Showing that the horseshoe set is locally minimal
I'm trying to prove the Smale's horseshoe set is locally minimal.
More specifically, let $H$ be the horseshoe set described in Section 1.8 in the book "Introduction to Dynamical Systems" by ...
- 794
8
votes
1
answer
362
views
Existence of Topologically Transitive Maps on nice Metric Spaces
Let $(X,d)$ be a separable metric space with no isolated points. Recall that a continuous function $T:(X,d)\rightarrow (X,d)$ is called topologically transitive if
Given $U,V$ non-empty open subsets ...
user683848
8
votes
1
answer
3k
views
Meaning of the term "topologically mixing"
I understand that for a system to behave chaotically, it needs to be "topologically mixing". However, I am not sure what that term really means.
There are several explanations of this online. ...
- 323
53
votes
1
answer
2k
views
Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
Let $a_0=1,a_n=\tan{a_{n-1}}$. Then is $\{a_n\}_{n=0}^\infty$ dense in $\Bbb{R}$?
I've drawn a map of this dynamical system and it seems that the sequence is dense on $\Bbb{R}$.
- 847