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Combination Laws for Scaling Exponents and Relation to the Geometry of Renormalization Operators

The Principle of Approximate Combination of Scaling Exponents

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Abstract

Renormalization group has become a standard tool for describing universal properties of different routes to chaos—period-doubling in unimodal maps, quasiperiodic transitions in circle maps, dynamics on the boundaries of Siegel disks, destruction of invariant circles of area-preserving twist maps, and others. The universal scaling exponents for each route are related to the properties of the corresponding renormalization operators.

We propose a Principle of Approximate Combination of Scaling Exponents (PACSE) that organizes the scaling exponents for different transitions to chaos. Roughly speaking, if the combinatorics of a transition is a composition of two simpler combinatorics, then the scaling exponents of the combined combinatorics is approximately equal to the product of the scaling exponents, both in the parameter space and in the configuration space, corresponding to each of these two combinatorics. We state PACSE quantitatively as precise asymptotics of the scaling exponents for combined combinatorics, and give convincing numerical evidence for it for each of the four dynamical systems mentioned above.

We propose an explanation of PACSE in terms of the dynamical properties of the renormalization operators—in particular, as a consequence of certain transversal intersections of the stable and unstable manifolds of the operators corresponding to different transition to chaos.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX, 78712, USA

    Rafael de la Llave

  2. IIMAS-UNAM, FENOMEC, Apdo. Postal 20-726, México, D.F., Mexico

    Arturo Olvera

  3. Department of Mathematics, University of Oklahoma, 601 Elm Avenue, Norman, OK, 73019, USA

    Nikola P. Petrov

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  1. Rafael de la Llave
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  2. Arturo Olvera
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  3. Nikola P. Petrov
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Correspondence to Nikola P. Petrov.

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The research of RL has been partially supported by NSF grant DMS 0901389, and Texas Coordinating Board ARP 0223. The research of NPP has been partially supported by NSF grant DMS 0807658. RL and NPP were also supported by Big XII fellowships. The computations were carried out on the computers of the Department of Mathematics of the University of Texas and the computers of IIMAS-UNAM.

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de la Llave, R., Olvera, A. & Petrov, N.P. Combination Laws for Scaling Exponents and Relation to the Geometry of Renormalization Operators. J Stat Phys 143, 889–920 (2011). https://doi.org/10.1007/s10955-011-0211-1

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