Thermodynamics of chaotic scattering at abrupt bifurcations

Thermodynamic and multifractal spectra of chaotic scattering are investigated at abrupt bifurcations leading from fully developed chaotic to regular scattering as the particle energy E passes through a critical value Eₘ. For processes with uniform scaling, the spectra of Lyapunov exponents, entropies, and partial dimensions diverge, stay constant, and vanish, respectively, when the energy approaches Eₘ from below. To characterize processes also having scattering angles close to 90^∘{}, a new quantity is introduced, the partial topological entropy depending on the concentration of single scatterings with angles close to 90^∘{}. Multifractal spectra are proved to be universal in the sense that they depend on topological properties only. In particular, the generalized dimensions scale as Dq=dq|ln(Eₘ-E)| for q of order 1|ln(Eₘ-E)|, otherwise, Dq is finite and zero in the negative and positive q ranges, respectively. Our approach thus justifies a conjecture of Bleher, Ott, and Grebogi [Phys. Rev. Lett. 63, 919 (1989)] for $q=0$, and yields explicit expressions for the coefficients dq. Phase transitions arising at the bifurcation point are analyzed.