# On the structure of isentropes of polynomial maps

@article{Bruin2013OnTS,
title={On the structure of isentropes of polynomial maps},
author={Henk Bruin and Sebastian van Strien},
journal={Dynamical Systems},
year={2013},
volume={28},
pages={381 - 392}
}
• Published 3 March 2013
• Mathematics
• Dynamical Systems
The structure of isentropes (i.e. sets of constant topological entropy) including the monotonicity of entropy has been studied for polynomial interval maps since the 1980s. We show that isentropes of multimodal polynomial families need not be locally connected and that entropy does in general not depend monotonically on a single critical value.
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#### References

SHOWING 1-10 OF 35 REFERENCES
On Entropy and Monotonicity for Real Cubic Maps
• Mathematics
• 1998
Abstract:Consider real cubic maps of the interval onto itself, either with positive or with negative leading coefficient. This paper completes the proof of the “monotonicity conjecture”, whichExpand
Monotonicity of entropy for real multimodal maps
• Mathematics
• 2009
In [16], Milnor posed the Monotonicity Conjecture that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected.Expand
On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II
The results about measures with maximal entropy, which are proved in [3], are extended to the following more general class of transformations on the unit intervalI : I=∪i=1/nJi, theJi are disjointExpand
Remarks on Iterated Cubic Maps
• J. Milnor
• Mathematics, Computer Science
• Exp. Math.
• 1992
This articl e discusses the dynamics of iterated cubic maps from the real or complex line to itself and will describe the geography of the parameter space for such maps. It is a rough survey with fewExpand
Topological Entropy of Unimodal Maps
In Section 1, we give the definition and general properties of the topological entropy of a map f : X→ X where X is a compact (metrizable) space.
The Connected Isentropes Conjecture in a Space of Quartic Polynomials
This note is a shortened version of my dissertation paper, defended at Stony Brook University in December 2004. It illustrates how dynamic complexity of a system evolves under deformations. The Expand
Rigidity for real polynomials
• Mathematics
• 2007
We prove the topological (or combinatorial) rigidity property for real polynomials with all critical points real and nondegenerate, which completes the last step in solving the density of Axiom AExpand
A horseshoe for the doubling operator: Topological dynamics for metric universality
• Physics
• 1987
Abstract The doubling operator, properly defined on the space of smooth maps on the interval at the boundary of chaos, yields a dynamical system in this function space. Even if one restricts oneselfExpand
Multipliers of periodic orbits in spaces of rational maps
• G. Levin
• Mathematics
• Ergodic Theory and Dynamical Systems
• 2010
Abstract Given a polynomial or a rational function f we include it in a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. ThenExpand
A Cr unimodal map with an arbitrary fast growth of the number of periodic points
• Mathematics
• Ergodic Theory and Dynamical Systems
• 2011
Abstract In this paper we present a surprising example of a Cr unimodal map of an interval f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any ahead given sequence along aExpand