arXiv subjects

Michael Benedicks

Publications and source records attributed to Michael Benedicks.

8 recordsLinked to original sources

arXiv2026

Topological Entropy for Power-Law Unimodal Maps

In this paper we prove that the monotonicity of kneading sequences and topological entropy, a fundamental structural property of the quadratic family, extends to the class of power-law unimodal maps $f_a(x)=a-|x|^r$ for arbitrary critical exponent $r>1$. This generalization is nontrivial: the absence of polynomial structure and the presence of non-integer criticality preclude the direct use of classical arguments. Our approach adapts and extends the Milnor-Thurston framework by introducing a Thurston-type operator associated with the critical orbit and establishing a determinant identity that relates its linearization to the parameter derivative of the orbit. The main difficulty proving positivity of this determinant in the absence of algebraic structure - is resolved via a contraction argument on an associated Torelli space endowed with the Teichm\"uller metric, extending Thurston's pullback construction beyond the polynomial setting, that is to critical powers $r=2^\nu/k$, $\nu\geq 1$, $k$ odd, and finally use continuity in $r$. As a consequence, we show that the kneading sequence varies monotonically with the parameter, and hence that the topological entropy is an increasing function of $a$. Our results show that the combinatorial organization of parameter space familiar from the quadratic family persists for unimodal maps with arbitrary power-law criticality, indicating that monotonicity of entropy is a robust phenomenon beyond polynomial dynamics.

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arXiv2020

Expansion properties of Double Standard Maps

For the family of Double Standard Maps $f_{a,b}=2x+a+\frac{b}{\pi} \sin2\pi x \quad\pmod{1}$ we investigate the structure of the space of parameters $a$ when $b=1$ and when $b\in[0,1)$. In the first case the maps have a critical point, but for a set of parameters $E_1$ of positive Lebesgue measure there is an invariant absolutely continuous measure for $f_{a,1}$. In the second case there is an open nonempty set $E_b$ of parameters for which the map $f_{a,b}$ is expanding. We show that as $b\nearrow 1$, the set $E_b$ accumulates on many points of $E_1$ in a regular way from the measure point of view.

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arXiv2019

Newhouse Laminations

Newhouse laminations occur in unfoldings of rank-one homoclinic tangencies. Namely, in these unfoldings, there exist codimension $2$ laminations of maps with infinitely many sinks which move simultaneously along the leaves. As consequence, in the space of real polynomial maps, there are examples of: Hénon maps, in any dimension, with infinitely many sinks, quadratic Hénon-like maps with infinitely many sinks and a period doubling attractor, quadratic Hénon-like maps with infinitely many sinks and a strange attractor, non trivial analytic families of polynomial maps with infinitely many sinks.

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arXiv2018

Coexistence phenomena in the H\'enon family

We study the classical H\'enon family $f_{a,b}:(x,y)\mapsto(1-ax^2+y,bx)$, $0<a<2$, $0<b<1$, and prove that given an integer $k\geq 1$, there is a set of parameters $E_k$ of positive two-dimensional Lebesgue measure so that $f_{a,b}$, for $(a,b)\in E_k$, has at least $k$ attractive periodic orbits and one strange attractor. A corresponding statement also holds for the H\'enon-like families. The final main result of the paper is the existence, within the classical H\'enon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.

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arXiv2017

Unique Continuation for the Magnetic Schr\"odinger Equation

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schr\"odinger equation. This property guarantees that if a solution of the Schr\"odinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry.

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arXiv2014

Non-existence of a Hohenberg-Kohn Variational Principle in Total Current Density Functional Theory

For a many-electron system, whether the particle density $ρ(\mathbf{r})$ and the total current density $\mathbf{j}(\mathbf{r})$ are sufficient to determine the one-body potential $V(\mathbf{r})$ and vector potential $\mathbf{A}(\mathbf{r})$, is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional $\mathord{\cal E}_{V_0,\mathbf{A}_0}(ρ,\mathbf{j}) = \langle ψ(ρ,\mathbf{j}),H(V_0,\mathbf{A}_0)ψ(ρ,\mathbf{j})\rangle$ can be minimal for densities that are not the ground-state densities of the fixed potentials $V_0$ and $\mathbf{A}_0$. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $ρ$ and $\mathbf{j}$, we show that a variational principle for Total Current Density Functional Theory as that of Hohenberg-Kohn for Density Functional Theory does not exist. The reason is that the assumed map from densities to the vector potential, written $(ρ,\mathbf{j})\mapsto \mathbf{A}(ρ,\mathbf{j};\mathbf{r})$, enters explicitly in $\mathord{\cal E}_{V_0,\mathbf{A}_0}(ρ,\mathbf{j})$.

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arXiv2013

Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps

We consider C^2 families t->f_t of C^4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure m_t of f_t, as a function of t on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set D of CE parameters, and, for each s in D, a subset D0 of D of polynomially recurrent parameters containing s as a Lebesgue density point, and constants C>1, G >4, so that, for every 1/2-Holder function A (of 1/2-Holder norm |A|) and all t in D0, |\int A dm_t -\int A dm_s| < C |A| |t-s|^{1/2} |log|t-s||^G (If f_t(x)=tx(1-x), the set D contains almost all CE parameters.) Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter s, we find a set of CE parameters D' accumulating at s, a constant C >1, and an infinitely differentiable function B, so that for all t in D' C |t-s|^{1/2} > |\int B dm_t -\int B dm_s| > |t-s|^{1/2}/C

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arXiv2003

Non uniformly hyperbolic dynamics: Hénon maps and related dynamical systems

In the 1960s and 1970s a large part of the theory of dynamical systems concerned the case of uniformly hyperbolic or Axiom A dynamical system and abstract ergodic theory of smooth dynamical systems. However since around 1980 an emphasize has been on concrete examples of one-dimensional dynamical systems with abundance of chaotic behavior (Collet &Eckmann and Jakobson). New proofs of Jakobson's one-dimensional results were given by Benedicks and Carleson \cite{BC85} and were considerably extended to apply to the case of Hénon maps by the same authors \cite{BC91}. Since then there has been a considerable development of these techniques and the methods have been extended to the ergodic theory and also to other dynamical systems (work by Viana, Young, Benedicks and many others). In the cases when it applies one can now say that this theory is now almost as complete as the Axiom A theory.

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