Is there a compact metric space XX and a continuous function f:X→Xf:X\to X such that the dynamic system (X,f)(X, f) is minimal, topologically mixing, and not expansive?

I took a look at tons of examples but couldn’t find such a system.

Some thoughts:

XX can’t be an interval of real numbers (else ff can’t be minimal).

(X,f)(X,f) can’t be a subshift (else it’s expansive since the full shift is).

XX must be perfect, i.e. have no isolated points (else ff isn’t sensitive and in particular isn’t topologically mixing).

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Circle homeomorphisms are never expansive, so maybe that would be a good place to look. However the obvious candidate for a minimal action, irrational rotations, is not topologically mixing.

– Dan Rust

2 days ago

@DanRust: thanks for looking into it. Do you think this could be relevant: webusers.imj-prg.fr/~bassam.fayad/smb.pdf ?

– Leo

2 days ago

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