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 $j$ ...
Nik Weaver's user avatar
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 ...
Ben McKay's user avatar
  • 25.3k
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 ...
HenrikRüping's user avatar
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,...
Alexandre Eremenko's user avatar
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\...
C7X's user avatar
  • 1,110
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. ...
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)...
Fedor Petrov's user avatar
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....
Vaughn Climenhaga's user avatar
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 ...
Damien's user avatar
  • 146
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, ...
Ronnie Pavlov's user avatar
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 ...
John Dawkins's user avatar
  • 1,799
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+...
Max Alekseyev's user avatar
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 ...
loup blanc's user avatar
  • 3,564

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