The dimensions of a non-conformal repeller and an average conformal repeller

Abstract

Abstract. In this paper, using thermodynamic formalism for the sub-additive potential, upper bounds for the Hausdorff dimension and the box dimension of non-conformal repellers are obtained as the sub-additive Bowen equation. The map f only needs to be C 1 , without additional conditions. We also prove that all the upper bounds for the Hausdorff dimension obtained in earlier papers coincide. This unifies their results. Furthermore we define an average conformal repeller and prove that the dimension of an average conformal repeller equals the unique root of the sub-additive Bowen equation.

The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials

“…The set J is called an average conformal repeller if for any T -invariant ergodic measure µ, λ 1 (µ) = λ 2 (µ) = · · · = λ m (µ) > 0, where λ i (µ), 1 ≤ i ≤ m denotes the Lyapunov exponent of T with respect to µ. This notion is introduced in [16], and the authors showed that Since the function sequence {log ‖Df n (x)‖} n≥1 is sub-additive and {log m(Df n (x))} n≥1 is sup-additive. Then, it is easy to see that the potential F = {log ‖Df n (x)‖} or F = {log m(Df n (x))} is asymptotically additive.…”

“…The potential F = {log ‖Df n (x)‖} or F = {log m(Df n (x))} may be not almost-additive. We just remark that the dimension of average conformal repeller can be computed by the zero of the sub-additive topological pressure, see [16] for details. And authors have proved that the Hausdorff dimension of average conformal repeller is stable under random perturbation, see [17] for details.…”

The local variational principle of topological pressure for sub-additive potentials

“…The topological pressure for a nonadditive sequence of potentials has proved a valuable tool in the study of the multifractal formalism of dimension theory, especially for nonconformal dynamical systems [15,12,16,11]. In [12,11] the authors use the sub-additive topological pressure to give an upper bound estimate of the Hausdorff dimension for the nonconformal repeller.…”

“…In [12,11] the authors use the sub-additive topological pressure to give an upper bound estimate of the Hausdorff dimension for the nonconformal repeller. In [15,16], the authors use the variational principle for the sub-additive topological pressure to give an upper bound estimate of the Hausdorff dimension for the nonconformal repeller and obtain the Hausdorff dimension for average conformal repellers as the root of the Bowen equation for the sub-additive topological pressure.…”

Dimensional Characteristics of the Nonwandering Sets of Open Billiards

“…The concept of an average conformal repeller was developed as a generalisation of a conformal repeller [1]. We extend this idea by generalising conformal hyperbolic sets to average conformal hyperbolic sets.…”