12 problems

For a real number $\beta>1$, let $T_{-\beta}$ denote the negative $\beta$-transformation and let its invariant measures be defined analogously to those of the usual $\beta$-transfo…

Let $\mathcal{T}$ be the parameter triangle for the family of beta-transformations, with parameters $(\alpha,\beta)$. A beta-transformation is topologically LEO when it has the loc…

Let $\mathcal{T}$ be the parameter triangle for the family of beta-transformations, with parameters $(\alpha,\beta)$. Transitivity conjecture. For all parameters in $\mathcal{T}$ s…

Let $M(\beta)$ denote the modulus of the second eigenvalue or eigenvalues of the Perron–Frobenius operator for the beta-map with parameter $\beta$. Flatto–Lagarias–Poonen conjectur…

Kalle–Kong–Langeveld–Li conjecture. For every $\beta\in(1,2]$ and every $t\in(0,1)$,

A generalized $\beta$-transformation has parameters $\beta>1$ and $\alpha\in[0,1]$. It has matching if there exists a finite integer $n$ such that … The smallest such $n$ is called…

Let $E_\infty$ be the set of bases defined in the paper, and let $\beta$ be a base in $E_\infty$. Transcendence conjecture. Each base … is transcendental. The conjecture is motivat…

Let $T\colon [0,1)\to[0,1)$ be the dynamical system defined by the composition $T_{\beta_{n-1}}\circ\cdots\circ T_{\beta_0}$, and let $S$ be the matrix whose entries are … The cond…

Let $\beta\in(1,2]$. The bifurcation-set equality conjecture. … if and only if $\mathcal{E}_\beta$ has no isolated points. Here $\mathcal{E}_\beta$ is the set-valued bifurcation se…

A multinacci number is a Pisot number that is the real root greater than $1$ of $\beta^n=\beta^{n-1}+\cdots+\beta+1$ for some integer $n\geq 2$. For such a $\beta$, let $T_\alpha$…

Let $\phi\in(0,\pi)$, and let $T_\phi$ be a $\phi$-optimal function, meaning that $T_\phi(\lambda_\phi e^{i\phi})=0$. The function $T_\phi$ is unique when $\phi/2\pi$ is irrational…

For $\beta>1$, let $T_{-\beta}$ denote the negative beta transformation on its natural interval, and let exactness mean that every positive-measure measurable set has iterates whos…