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(1) For $\beta>1$ let $T_{\beta}:[0,1)\to [0,1)$ be given by $T_{\beta}(x)=\{\beta x\}$, where $\{\cdot\}$ denotes the non-integer part of real number. It is classical result (due to Parry I think) th …

Let $(X,T)$ and $(Y,G)$ be topological dynamical systems. If $(Y,G)$ is a factor of $(X,T)$ it is well known and easy to proof that $h(G)\le h(T)$ , where $h$ denotes the topological entropy. If the s …

I am searching for dynamical systems on compact spaces which are Devaney chaotic but have topological entropy zero. On the interval such systems do not exist. I think on the Cantor space and on the to …

Is there a notion of dimension such that for all Borel sets $A,B\subseteq\mathbb{R}^{n}$ we have $$ \dim(A\times B)=\dim(A)+\dim(B)?$$ For topological, Minkowski, packing and Hausdorff dimension this …

Katok / Hasselblatt: Introduction to the Modern Theory of Dynamical Systems gives a good introduction.

Before asking my question I like to admit that i am not an expert in the foundation of mathematics and I am interested in this issue more from a philosophical perspective. So my question may be comp …

Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into half-open intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose …

Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?

Two mathematicians from North Korea I know are Kim, Jinhyon and Ju, Hyonhui. I like their paper: Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coup …

Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have thi …

Starting to write an introduction to the philosophy of mathematics, I find tons of positions that are of historical interest. Which philosophical positions are explicitly considered these days, say in …

I find many numerical results on the three-body problem, but what is rigorously proved? Especially I would be interested in the parameter domains for which we have rigorous lower bounds on the topolog …

Let $(a_{i})$ be an increasing sequence of positive integers given by a linear recurrence $a_{i+n}=c_{n}a_{i+n-1}+\dots +c_{1}a_{i}$ with $c_{i}\in\{-1,0,1\}$ and $a_{i}=2^{i}$ for $i=1,\dots n$ such …

Let $(a_i)$ be an exponential increasing sequences of natural numbers; there are constants $a\in(1,2)$, $b>0$ such that $|a_{i}-ba^i|$ is exponentially decreasing. Let $(s_{k})$ be a sequences in $\{- …

The book of Tabachnikov https://www.math.psu.edu/tabachni/Books/billiardsgeometry.pdf is fine as an introducion to billiards