Monotonicity of entropy for unimodal real quadratic rational maps

Yan Gao

doi:10.3934/dcds.2022101

Article Contents

2022, Volume 42, Issue 11: 5377-5386. Doi: 10.3934/dcds.2022101

This issue Previous Article Mingled hyperbolicities: Ergodic properties and bifurcation phenomena (an approach using concavity) Next Article An

$ L^{q}\rightarrow L^{r} $

estimate for rough Fourier integral operators and its applications

Monotonicity of entropy for unimodal real quadratic rational maps

Yan Gao^,

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, China

^*Corresponding author: Yan Gao
^*Corresponding author: Yan Gao

Received: September 2021

Revised: April 2022

Early access: August 2022

Published: November 2022

The author is supported by NSFC grants 11871354 and 12131016.

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Abstract

We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients and real critical points. This confirms a conjecture made in [8].

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Mathematics Subject Classification: Primary: 37B40, 37F10, 37F20.

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Figure 1. An entropy contour plot for the unimodal quadratic rational maps 8 adapted from [8]. The ordering of colors is black$ < $blue$ < $magenta$ < $green$ < $cyan$ < $yellow$ < $red and they correspond to the partition $ [0, 0.1) $, $ [0.1, 0.25) $, $ [0.25, 0.4) $, $ [0.4, 0.48) $, $ [0.48, 0.55) $, $ [0.55, 0.65) $ and $ [0.65, \log(2)] $ of $ [0, \log(2)\approx 0.7] $. The connectedness of entropy level sets here suggests the monotonicity of entropy in the unimodal region of $ {\rm rat}_2({\mathbb R}) $

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Figure 2. A schematic picture of bones (in black) defined by the post-critical relations $ f^{ n}(c_1) = c_1 $ in the unimodal region of the $ (\sigma_1, \sigma_2) $-plane. They are disjoint one-dimensional submanifolds. There are bone-arcs connecting PCF quadratic polynomials to PCF quadratic rational maps on the $ f(c_1) = c_2 $ line. Any other bone in this region should be a Jordan curve, a bone-loop. The picture is adapted from [8]

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