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By a potential space filling curve we mean a continuous function $f:[0,1]\to [0,1]$ such that there is a continuous surgective function $g:[0,1]\to [0,1]^2$ with $f=\pi_1 \circ g$ where $\pi_1$ is the projection on the first coordinate. I am curious about the topological entropy of a typical potential space filling curve $f:[0,1]\to [0,1]$.

Are there both examples of vanishing and non vanishing of such potential filling curve?

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    $\begingroup$ Is there a map that you are considering applying? $\endgroup$
    – Anthony Quas
    Commented Jul 13 at 0:48
  • $\begingroup$ @AnthonyQuas Thank you for your comment. In fact by this post I am curious about the diversity of space filling curve from the entropy point of view. What quantities can be realized as the entropy of some potential filling curve? Can one say that the entropy is always non zero? $\endgroup$
    – Ali Taghavi
    Commented Jul 13 at 10:18
  • $\begingroup$ I guess my comment is that I only know how to define entropy when there is a dynamical system. In your question, you have a space, but I don’t see the dynamics. $\endgroup$
    – Anthony Quas
    Commented Jul 14 at 2:14
  • $\begingroup$ @AnthonyQuas I think topologicql entropy is defined for non injective maps too. Here $f$ is a map on the interval which is composition of the projection $\pi_2$ with a space filling curve. so we have a dynamic on the interval $\endgroup$
    – Ali Taghavi
    Commented Jul 14 at 8:47
  • $\begingroup$ @AnthonyQuas is injectivity of f the point you are indicating to?or you meqn consideration of a precise example of $f$? $\endgroup$
    – Ali Taghavi
    Commented Jul 14 at 8:54

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