Papers by Charles Tresser
arXiv (Cornell University), Apr 2, 1996
We de ne the Farey web | a collection of loci in the parameter plane of families of simple non-in... more We de ne the Farey web | a collection of loci in the parameter plane of families of simple non-invertible maps of the circle. We prove some results about the arrangement of these loci and their relationships with other dynamically signi cant features of the parameter plane. The results enable us to provide short proofs for a number of theorems about the organization of frequency-locking.
arXiv (Cornell University), Sep 4, 2017
In the family of area-contracting Hénon-like maps with zero topological entropy we show that ther... more In the family of area-contracting Hénon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting Hénon-like maps in a family with finitely many parameters. A similar result, but for the chaotic maps in the family, became part of the folklore a short time after Hénon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractor in R 2. Our proof uses recent results about infinitely renormalisable area-contracting Hénon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Hénon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to {1, 2,. .. , 2 n−1 } or an initial segment of this n-tuple) increases with the number of periods. In comparison, among C k-embeddings of the 2-disk with k ≥ 1, the maximal moduli number for non-chaotic but non area-contracting maps in the interior of the set of zero-entropy is infinite.
Ergodic Theory and Dynamical Systems, Apr 1, 2009
The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet an... more The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser in the nineteen-seventieth to study the asymptotic small scale geometry of the attractor of one-dimensional systems which are at the transition from simple to chaotic dynamics. This geometry turns out to not depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point which is also hyperbolic among generic smooth enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that in the space of C 2+α unimodal maps, for α close to one, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main results states that in the space of C 2 unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to get a priori bounds. In this smoother class, called C 2+|•| the failure of hyperbolicity is tamer than in C 2. Things get much worse with just a bit less of smoothness than C 2 as then even the uniqueness is lost and other asymptotic behavior become possible. We show that the period doubling renormalization operator acting on the space of C 1+Lip unimodal maps has infinite topological entropy.
Physical Review Letters, Jan 18, 1993
On-off' intermittency is an aperiodic switching between static, or laminar, behavior and chaotic ... more On-off' intermittency is an aperiodic switching between static, or laminar, behavior and chaotic bursts of oscillation. It can be generated by systems having an unstable invariant (or quasi-invariant) manifold, within which is found a suitable attractor. We clarify the roles of such attractors in producing intermittency, provide examples, and relate them to previous work.
Proceedings of the American Mathematical Society, 1996
We prove that for continuous maps on the interval, the existence of an n-cycle implies the existe... more We prove that for continuous maps on the interval, the existence of an n-cycle implies the existence of n − 1 points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.
Communications in Mathematical Physics, Nov 1, 1992
In the first part of this paper, for each d > 2, we construct diffeomorphisms of the d-dimensiona... more In the first part of this paper, for each d > 2, we construct diffeomorphisms of the d-dimensional ball which have zero entropy, one periodic orbit with period 2 n for each n > 0, no other periodic orbits, and a single invariant Cantor set which has a continuum of possible but, in any case, simple geometric structures. These diffeomorphisms are C r(d)-smooth, where r(d) is a strictly increasing function of d, which goes to infinity with d. The second part contains a more general result about smooth maps obtained by an infinite sequence of surgeries, and further particular cases.
Physics Letters, Oct 1, 1984
We describe a new method for obtaining a quantitatrce measure of chaos in some maps and flows. Th... more We describe a new method for obtaining a quantitatrce measure of chaos in some maps and flows. This method is probably simple enough to be useful experimentally.
Proceedings of The Royal Society B: Biological Sciences, Feb 7, 1999
arXiv (Cornell University), Mar 5, 1994
We show that the conjugacy class of an eventually expanding continuous piecewise affine interval ... more We show that the conjugacy class of an eventually expanding continuous piecewise affine interval map is contained in a smooth codimension 1 submanifold of parameter space. In particular conjugacy classes have empty interior. This is based on a study of the relation between induced Markov maps and ergodic theoretical behavior.
arXiv (Cornell University), Nov 24, 2013
We prove a version of Bell's Theorem that does not assume locality. Beside quantum mechanics and ... more We prove a version of Bell's Theorem that does not assume locality. Beside quantum mechanics and weak forms of relativistic causality and of realism, we only assume one correlation to be twice differentiable at zero. Since we prove that said correlation is once differentiable at zero, this brings us one degree of differentiability away from a relativistic von Neumann no hidden variables theorem.
arXiv (Cornell University), Sep 7, 2017
For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is alw... more For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with respect to either the Hölder or the Sobolev topologies, topological entropy is generically infinite. We also prove versions of the C 1-Closing Lemma in either of these spaces. Finally, we give examples of homeomorphisms with infinite topological entropy which are Hölder and/or Sobolev of every exponent.
arXiv (Cornell University), Nov 6, 1995
We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the ... more We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point, i.e., the combinatorial description of the boundary of chaos coincides with the topological description. We also show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable.
arXiv (Cornell University), Jun 13, 2014
It is well known that different preparations of a mixed state cannot be distinguished by a measur... more It is well known that different preparations of a mixed state cannot be distinguished by a measurement of that state. Yet we show that some other experiments let us make this distinction despite a very general belief that this would not be possible. Issues in quantum cryptography that prompted this work are only briefly mentioned in this letter.
arXiv (Cornell University), Mar 1, 2005
In Bohm's version of the EPR gedanken experiment, the spin of the second particle along any vecto... more In Bohm's version of the EPR gedanken experiment, the spin of the second particle along any vector is minus the spin of the other particle along the same vector. It seems that either the choice of vector along which one projects the spin of the first particle influences at superluminal speed the state of the second particle, or naive realism holds true (i.e., the projections of the spin of any EPR particle along all the vectors are determined before any measurement occurs). Naive realism is negated by Bell's theory that originated and is still most often presented as related to non-locality, a relation whose necessity has recently been proven to be false. I advocate here that the solution of the apparent paradox lies in the fact that the spin of the second particle is determined along any vector, but not along all vectors. Such an any-all distinction was already present in quantum mechanics, for instance in the fact that the spin can be measured along any vector but not at once along all vectors, as a result of the Uncertainty Principle. The time symmetry of the any-all distinction defended here is in fact reminiscent of (and I claim, due to) the time symmetry of the Uncertainty Principle described by Einstein, Tolman, and Podolsky in 1931, in a paper entitled "Knowledge of Past and Future in Quantum Mechanics" that is enough to negate naive realism and to hint at the any-all distinction. A simple classical model is next built, which captures aspects of the any-all distinction: the goal is of course not to have a classical exact model, but to provide a caricature that might help some people.
arXiv (Cornell University), Jul 3, 2013
We investigate the so-called persistence problem of Sloane, exploiting connections with the dynam... more We investigate the so-called persistence problem of Sloane, exploiting connections with the dynamics of circle maps and the ergodic theory of Z d actions. We also formulate a conjecture concerning the asymptotic distribution of digits in long products of finitely many primes whose truth would, in particular, solve the persistence problem. The heuristics that we propose to complement our numerical studies can be thought in terms of a simple model in statistical mechanics.
Journal de physique. Lettres, 1984
2014 De nombreux flots bi-périodiques peuvent être modélisés par des applications du cercle dans ... more 2014 De nombreux flots bi-périodiques peuvent être modélisés par des applications du cercle dans lui-même. Nous décomposons en quatre composantes, la frontière du chaos topologique dans l'espace des applications C1 de degré 1 du cercle, et nous décrivons les routes typiques vers le chaos topologique. Abstract. 2014 Many biperiodic flows can be modelled by maps of a circle to itself. We decompose the boundary of topological chaos in the space of C1 circle maps of degree 1 into four subsets, and describe the typical routes to topological chaos.
Mathematical proceedings of the Cambridge Philosophical Society, Nov 1, 1984
We define two notions of order for periodic orbits of circle maps, and describe some of the conse... more We define two notions of order for periodic orbits of circle maps, and describe some of the consequences of possession of a badly ordered periodic orbit.
Annali della Scuola normale superiore di Pisa. Classe di scienze, Jun 24, 2021
For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is alw... more For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with respect to either the Hölder or the Sobolev topologies, topological entropy is generically infinite. We also prove versions of the C 1-Closing Lemma in either of these spaces. Finally, we give examples of homeomorphisms with infinite topological entropy which are Hölder and/or Sobolev of every exponent.
arXiv (Cornell University), Jan 6, 2005
We prove here a version of Bell's Theorem that is simpler than any previous one. The contradictio... more We prove here a version of Bell's Theorem that is simpler than any previous one. The contradiction of Bell's inequality with Quantum Mechanics in the new version is not cured by non-locality so that this version allows one to single out classical realism, and not locality, as the common source of all false inequalities of Bell's type.
arXiv (Cornell University), 2008
Two typical entanglements will be shown to stand on opposite sides on the issue of instrumental r... more Two typical entanglements will be shown to stand on opposite sides on the issue of instrumental realism, the issue of whether (as for EPR in the original form or EPRB, Bohm's versions using spin) or not (as for GHZ) observables have values that preexist measurement. Instrumental realism in the EPR context helps us prove that in some special circumstances, one can get simultaneous knowledge of two conjugate quantities, which in particular make sense together. This shatters the axiomatic presentation of Quantum Mechanics. This simultaneous knowledge of two conjugate quantities is an elaboration on 1935 work by Schrödinger, hence the name, Schrödinger Unorthodoxy Theorem, given to the second main result that is obtained with little effort from the result on instrumental realism. Once the axiomatic edifice of Quantum Mechanics is broken, we can let go the completeness of the wave function as suggested in EPR, and like Einstein "absolutely hold fast" on locality. The EPR paper of mid-1935 gets here contrasted, from a new point of view, with a 1936 text by Einstein where Einstein avoids using counterfactuals. Counterfactuals get a precise definition and the corresponding concept is used all along, but may be new under an old name. We provide a short critical review of Bell's 1964 paper. Then, a small modification of arguments for the Schrödinger Unorthodoxy Theorem will let appear a simple conservation law, combined with the Malus Law, as the origin of the correlation in Bell's version of EPRB: this is our third main result. As the fourth main result, a last use of the concept of counterfactual yields the decay of geometry at small enough scale. This opens a new world of interpretation of aspects of Quantum Mechanics, aspects that range from measurement and the need of classical physics to views on what realism should mean in microphysics.
Uploads
Papers by Charles Tresser