# Is a set over which dynamics are topologically conjugate to a shift map on two symbols always repelling?

Consider the one-sided full shift map $$sigma$$ and the associated shift space of infinite sequences in two letters $${0,1}^mathbb{N}$$ on which the shift map acts, equipped with the usual metric. Also consider another metric space $$(X,d)$$, along with a continuous map $$f$$ that evolves the elements of $$X$$ (i.e. it describes dynamics on $$X$$).

Assume that the action of $$f$$ on a compact subset of $$X$$ is topologically conjugate to the one-sided full shift map $$sigma$$ on $${0,1}^mathbb{N}$$.

Is that compact subset always a repelling set of $$f$$ on $$X$$?

MOTIVATION: All of the examples I know of that are actually topological conjugates of the shift map/space cause the mapped shift space to be a repelling set, like the tent map for $$mu = 3$$, the $$10x text{ mod } 1$$ map, and the logistic map for $$r > 4$$, which makes the chaos essentially "invisible" to computers. The examples for which the chaos is "visible" do not properly conjugate the shift space onto the underlying set over which the dynamics take place—for example, the $$2x text{ mod } 1$$ map, the logistic map for $$r=4$$, and the tent map for $$r = 2$$. (You can check this by observing that in the latter, the shift space—which is compact—is mapped onto open sets and/or the mapping is not a homeomorphism at all.)

Apologies if the question is elementary—I’ll remove it if indicated to.

MathOverflow Asked by aghostinthefigures on January 26, 2021

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