Open Problems in dynamics and related fields

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Some open problems

(These questions are rather old, but their status was updated in October 2023)

These are some problems I've thought about and would like to see solved. Some of them are mine, some I'm just passing on.

• Are the periodic points dense in multidimensional, strongly irreducible SFTs?
  I recently showed that there exist aperiodic strongy irreducible subshifts in any dimension >1, which is slightly related (but the problem is open).


• Some questions about automorphisms of shifts of finite type.


• A question about genericity and approxiation in the space of connected systems.


• Is there a factor-universal cellular automaton?


• Is there a ratio ergodic theorem (Hopf's theorem) for actions of non-abelian gruops?
  Kieran Jarrett proved that one does exist for the Heisenberg groups).
  On the other hand I showed that some groups (even some abelian groups) do not admit ratio theorems.

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Some closed problems

• Some questions about recurrence and entropy in topological systems.
  I answered the first problem in this paper
  The last problem in the link above was answered for nilpotent groups by Huang, Jin and Ye.


• Can the divergence of Gibbs measures at zero temperature be observed at the level of finite marginals?
  Solved by Coronel and Rivera-Letelier in this paper (Appendix A)


• Can we estimate an eigenvalue(s) of a process by observing finite samples?
  Solved (under some regularity conditions) by Anthony Quas.


• Is the multidimension odd shift sofic?
  Solved in dimension 2 by Julien Cassaigne.

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Other problem sites in dynamics

• A survey of open problems rekated to Henon maps (January 2024).


• Some Open Problems in Dynamical Systems & Ergodic Theory from Katsiveli 2000, with occasional updates.


• The Stony Brook problem page.


• Mike Boyle's open problems in symbolic dynamics.


• Nikos Frantzikinakis's survey of open problems on non-conventional ergodic averages.


• Alex Gorodnik's page contains an open problems survey


• Anatole Katok's problem page, and an old survey of his containing open problems (many perhaps closed by now).


• Some problems on algebraic dynamics by Tom Ward.


• Oliver Knill's Open problem's in Hamiltonian dynamics


• Problems from the 3rd Pingree Park workshop.
  The first problem in the file was solved by Ville Salo.


• Problems from icmds2010 (in video!).

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Problem sites in probability (probably out of date)

• Itai Benjamini has some problems on his homepage (mostly in probability).


• Michel Talagrand's fiscal stimulus.


• Some problems in probability by David Aldous.

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Other sites

• The big ones from the Clay Institute.

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Mike Hochman's home