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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 …

Let $X$ be a set and $(T,\cdot)$ an abelian group. Is there a category of $T$-dynamical systems on $X$ which yields useful information about $X$ and $T$? More precisely, is there a category whose obj …

Lovely question! Sadly, the answer is "no" in the sense that the fixed point property is not homotopy-invariant even in the category of finite polyhedra. In fact, it is also not invariant under the op …

A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context. Let $\mathcal{C}$ be a category. …

The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest. Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ri …

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a maxima …

The following question came up while constructing delay embeddings of time series data. Consider an unknown topological space $X$ and an unknown continuous function $f:X \to X$. We are given a combin …

Pick up (almost) anything by Ethan Akin. I particularly recommend "The General Topology of Dynamical Systems" available on Amazon. Although it is somewhat older than what you indicate you are looking …

Not all interesting examples come from algebraic geometry and number theory -- your questions are fairly natural in many other settings. For instance, Conway's game of life qualifies as an answer to y …

The answer to the question as stated seems to be "no". Consider a three-element Markov partition $\mathcal{M} = \{A, B, C\}$ with directed edges $(A,B)$, $(B,C)$ and $(C,A)$. There is an obvious peri …

It's not clear when Ghys made the slides to which you have linked. The only date I could find in those was 1963 (referring to the Lorenz equations), which would make the bound on "recent" rather gener …

The dynamics are ergodic with respect to Lebesgue measure. See G. Hedlund, Fuchsian groups and mixtures, Ann. of Math. Volume 40, Number 2 (1939) 370-383, available here. If you prefer somethi …

Your setup defines set-valued dynamics on $X$. More precisely, you have $$X \stackrel{p}{\leftarrow} \overline{G} \stackrel{q}{\rightarrow} X$$ where $p$ and $q$ are the obvious projection maps. The s …

A few observations first: I assume in your motivating example that the coarseness parameter $\epsilon$ is smaller than $1$, otherwise you can't even tell that cylinder apart from a point. Anton's co …