Tagged Questions
3
votes
2answers
64 views
Show the map $f(x)=\frac12 (x+1/x)$ has an attractive fixed point in $(0,\infty)$
From numerical test, I know $x=1$ is an attractive fixed point of the function
$$
f(x)=\frac12 \left(x+\frac{1}{x}\right),
$$
on $(0,\infty)$.
Is there a way to prove it?
Since
$$
...
2
votes
1answer
48 views
Ergodic theory question about the support of a measure.
I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question:
...
2
votes
2answers
76 views
Entropy of a North South Transformation.
Let $f:\mathbb{S}^2\to\mathbb{S}^2$ be a continuous north south Transformation, in other words, the point $(0,0,1)$ is a global attractor for $f$ and $(0,0,-1)$ is a global attractor for $f^{-1}$. ...
5
votes
1answer
122 views
Entropy of a Linear Toral Automorphism
I'm trying to calculate the entropy of the Linear Toral Automorphism
induced by
$$f(x,y,z)=(x,y+x,y+z)$$
This is an exercise in the Katok book.
This map has all eigenvalues equal to 1. But I do ...
1
vote
1answer
57 views
Dynamical system: hypothesis on metric functions
Let $E$ be a completely metrizable separable topological space and $\mathscr E$ be its Borel $\sigma$-algebra. Consider a measurable map $F:E\to E$ such that
if $f:E\to \mathbb R$ is continuous and ...
1
vote
1answer
110 views
Dynamics Question
Let be $T_{\beta}:[0,1]\to [0,1]$ defined by $T_{\beta}(x)=\beta x \bmod 1$ where $\beta \in (1,2).$
Questions:
$T_{\beta}$ is topologically transitive?
What about the periodic points?
...
2
votes
1answer
37 views
Product of Transitive Systems
Let be $M$ a topological space, and $f:M\to M$ a danymical system, i.e, a continuous map between from $M$ to $M$.
We say that a dynamical system, $f:M\to M$ is topologically transitive when, ...
