All Questions

I need to estimate $|x - f^{-1}(g^{-1}(f(g(x))))|$ for various values of $x$ for two smooth invertible functions $f$ and $g$ on $\mathbb{R}$ (actually some other spaces, but $\mathbb{R}$ will do.) Are ...

This is related to discrete dynamical systems, with the initial condition $X_1$ being a random variable with a non singular distribution. The system is driven by the iteration $X_{n+1} = g(X_n)$ for ...

This is a weak version of this problem, written down in Lviv Scottish Book. I start with necessary definitions. Let $K=-K$ be a centrally symmetric compact convex body in the Euclidean space $\mathbb ...

Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem ...

Background At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...