1
vote
0answers
26 views

The future of the orbit of a point is a closed set [duplicate]

$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system. Prove: $w(x_{0})$ is closed. Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid ...
1
vote
0answers
62 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...
0
votes
0answers
54 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
4
votes
2answers
112 views

A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
0
votes
1answer
180 views

Bounded sets of isolated points in compact metric spaces

Context and definitions: Say $(X,d)$ is a compact metric space, with $f: X \rightarrow X$ continuous. For each $n \in \mathbb{N}$, the metric $d_{n}(x,y) = \max_{0 \leq k \leq ...
3
votes
2answers
94 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 $$ ...
3
votes
2answers
142 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: ...
3
votes
2answers
79 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
150 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
61 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
136 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
58 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, ...