Every minimal homeomorphism of a Cantor set is strongly orbit equivalent to a homeomorphism of infinite entropy.

A graph is called $K_{1,n}$-free if it contains no $K_{1,n}$ as an induced subgraph. Let $a$, $b$ $(0\leq a<b)$, and $n$ $(\geq 3)$ be integers. Let $G$ be a $K_{1,n}$-free graph. We prove that $G$ has an $[a,b]$-factor if its minimum degree is at least \[ \left(\frac{(a+1)(n-1)}{b}+1\right)\left\lceil\frac{a}{2}+\frac{b}{2(n-1)}\right\rceil-\frac{n-1}{b}\left(\left\lceil\frac{a}{2}+\frac{b}{2(n-1)}\right\rceil\right)^2-1. \] This degree condition is sharp for any integers $a$, $b$, and $n$ with $b\leq a(n-1)$. If $b\geq a(n-1)$, it exists if its minimum degree is at least $a$.

Every Morse-Smale diffeomorphism of the circle is conjugate to a diffeomorphism belonging to the set defined by \[ f_{\omega,\varepsilon,k}(x)=x+\omega+\frac{\varepsilon}{2\pi}\sin(2k\pi x) \quad (0<\omega<1, 0<\varepsilon<1, k \text{ with } 0<\varepsilon k<1) \] and Morse-Smale diffeomorphisms in the set is $C^1$ open and dense, with respect to the relative topology, in Amol'd tongue.

Certain piecewise linear expanding maps on a finite union of polygons in $\mathbf{R}^2$ are considered. The Perron-Frobenius operator associated with a map is considered on a locally convex linear space which is an extension of the space of bounded variation functions, and the spectrum of it is determined by Fredholm matrices. New signed symbolic dynamics are defined by using screens, and the Fredholm matrices are constructed by renewal equations on this signed symbolic dynamics.