No semiconjugacy to a map of constant slope

“…By Lemmas 5.2 and 5.3, the inequality (12) holds for every interval K. If K is a nondegenerate interval, then there is s such that f s (K) = X. Then, by (12), µ(K) > 0. Now we are ready to prove the existence of a fair measure.…”

Section: Piecewise Monotone Interval Mapsmentioning

confidence: 93%

“…The following lemma is well known, and is basically a special case of Lemma 4.1 of [12]. Notice that although formally in [12] the assumption is that the slope is constant, what is really used in the proof is only that it is larger than 2. Note also that in our terminology "piecewise" means piecewise with finitely many pieces.…”

Section: Now (14) Follows From (15) and (16)mentioning

confidence: 98%

Counting preimages

Misiurewicz¹,

Rodrigues²

2017

Ergod. Th. Dynam. Sys.

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Abstract. For noninvertible maps, mainly subshifts of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, random in the sense that at each step every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It tuns out that instead of the topological entropy we get the metric entropy of a special measure, which we call the fair measure. In general this entropy (the fair entropy) is smaller than the topological entropy. In such a way, for the systems that we consider, we get a new natural measure and a new invariant of topological conjugacy.

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“…Clearly f is countably Markov with partition set P . Moreover, f is topologically mixing as a consequence of [11,Theorem 5.4].…”

Section: An Examplementioning

confidence: 99%

The infimum of Lipschitz constants in the conjugacy class of an interval map

Bobok

Roth

2018

Proc. Amer. Math. Soc.

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How can we interpret the infimum of Lipschitz constants in a conjugacy class of interval maps? For positive entropy maps, the exponential of the topological entropy gives a well-known lower bound. We show that for piecewise monotone interval maps as well as for C ∞ interval maps, these two quantities are equal, but for countably piecewise monotone maps, the inequality can be strict. Moreover, in the topologically mixing and Markov case, we characterize the infimum of Lipschitz constants as the exponential of the Salama entropy of a certain reverse Markov chain associated with the map. Dynamically, this number represents the exponential growth rate of the number of iterated preimages of nearly any point.

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No semiconjugacy to a map of constant slope

Cited by 9 publications

References 12 publications

Topological dynamics of piecewise -affine maps

Topological dynamics of piecewise -affine maps

Counting preimages

The infimum of Lipschitz constants in the conjugacy class of an interval map

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