11
votes

Accepted

### If the average of a sequence converges, can I find a uniform bound that does not depend on where I start?

The sequence $1,0,1,1,0,0,1,1,1,0,0,0,\ldots$ is a counterexample. For each $j$ we have $\frac{1}{n}\sum_{k=1}^n a_{k+j} \to \frac{1}{2}$, but for any proposed $N_\epsilon$ we can find a value of $j$ ...

9
votes

### Is the geodesic flow on a Riemannian manifold conservative?

Consider a compact Riemannian manifold. Its geodesic flow preserves its unit sphere bundle, also compact. On the unit sphere bundle, any potential will have a minimum and a maximum, so there will be ...

7
votes

Accepted

### Criteria for extending vector field on sphere to ball

View $B^{n}$ as $(S^{n-1}\times[0,1])/(S^{n-1}\times \{1\})$. This means that an extension of a map from $B^{n}$ to somewhere is just an extension to $S^{n-1}\times [0,1]$ which is constant on the ...

4
votes

### Entire function of finite order with deficient value

The theorem you stated, and its various versions and generalizations, are the only simple sufficient conditions for $\delta(0)>0$.
For example, if $f$ is entire of genus $1$, and zeros lie on a ray,...

2
votes

### How can I catalog these generalized Collatz problems?

This is a ruleset whose termination is independent from $\mathrm{PRA}$, primitive recursive arithmetic, letting $\langle x,y\rangle$ be short for $(x+y)(x+y+1)+2y$:
$\langle y,\langle 0,z+1\rangle\...

2
votes

### Furstenberg $\times 2 \times 3$ conjecture, bibliography

After Rudoplph's and before the EKL paper came Johnson's in 1992:
Aimee S. A. Johnson. Measures on the circle invariant under multiplication by a nonlacunary
subsemigroup of the integers. Israel J. ...

Community wiki

2
votes

Accepted

### Functional equations based on composition

There are no solutions, even if we only assume that $f$ has a derivative at every point.
Note that if $f(x)>x$ for all $x$, then $f(f(x))>f(x)>x$ etc, thus $\sum a_k f^k(x)\geqslant (\sum a_k)...

2
votes

### If the average of a sequence converges, can I find a uniform bound that does not depend on where I start?

Regarding your original question about Birkhoff averages, the story is the following: Suppose $X$ is a compact metric space, $T\colon X\to X$ is continuous, and $f\colon X\to \mathbb{R}$ is continuous....

2
votes

Accepted

### Devaney chaos and topological entropy

An example of a dynamical system exhibiting Devaney chaos with zero topological entropy is constructed in the paper "Entropy and Exact Devaney Chaos on Totally Regular Continua". The key ...

1
vote

### Devaney chaos and topological entropy

It's fairly easy to make such a system symbolically (i.e. on the Cantor set), but I'm not sure of examples in print. Here is the simplest one I could think of.
Our system $(X, T)$ will be a subshift, ...

1
vote

Accepted

### Same occupation measure $\Rightarrow$ same trajectory

Under your hypotheses, you'll have (for each bounded continuous function $\varphi$)
$$
\int_0^T \varphi(x^f(s))\,ds = \int_0^T \varphi(x^g(s))\,ds,\qquad\forall T\ge 0,
$$
where $x^f$ and $x^g$ are ...

1
vote

### Functional equations based on composition

As it was pointed in the comments, the case of a linear function $px$ should be excluded. Let's investigate the given sum on a linear function $f(x)=px+q$ with $q\ne0$.
We have
$$\sum_{k=0}^n a_k (px+...

1
vote

### Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative

$\DeclareMathOperator\spectrum{spectrum}\DeclareMathOperator\GL{GL}$As Nathaniel wrote, the case $X\in M_n(\mathbb{Q})$ is not difficult.
Let $p>0$ be an integer s.t. $pX\in M_n(\mathbb{Z})$. The ...

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

ds.dynamical-systems × 2376ergodic-theory × 488

reference-request × 281

differential-equations × 214

dg.differential-geometry × 209

ca.classical-analysis-and-odes × 142

nt.number-theory × 131

symbolic-dynamics × 123

pr.probability × 115

fa.functional-analysis × 114

measure-theory × 86

riemannian-geometry × 84

complex-dynamics × 83

gn.general-topology × 81

mp.mathematical-physics × 79

gr.group-theory × 78

real-analysis × 74

foliations × 74

mg.metric-geometry × 67

gt.geometric-topology × 64

ap.analysis-of-pdes × 63

sg.symplectic-geometry × 63

cv.complex-variables × 58

discrete-dynamical-systems × 56

hyperbolic-dynamics × 54