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Long hitting times for expanding systems
- Source :
- Nonlinearity. 33:942-970
- Publication Year :
- 2020
- Publisher :
- IOP Publishing, 2020.
-
Abstract
- We prove a new result in the area of hitting time statistics. Currently, there is a lot of papers showing that the first entry times into cylinders or balls are often faster than the Birkhoff's Ergodic Theorem would suggest. We provide an opposite counterpart to these results by proving that the hitting times into shrinking balls are also often much larger than these theorems would suggest, by showing that for many dynamical systems $$ \displaystyle \limsup_{r\to 0} \tau_{B(y,r)}(x)\mu(B(y,r))=+\infty, $$ for an appropriately large, at least of full measure, set of points $y$ and $x$. We first do this for all transitive open distance expanding maps and Gibbs/equilibrium states of H\"older continuous potentials; in particular for all irreducible subshifts of finite type with a finite alphabet. Then we prove such result for all finitely irreducible subshifts of finite type with a countable alphabet and Gibbs/equilibrium states for H\"older continuous summable potentials. Next, we show that the \emph{limsup} result holds for all graph directed Markov systems (far going natural generalizations of iterated function systems) and projections of aforementioned Gibbs states on their limit sets. By utilizing the first return map techniques, we then prove the \emph{limsup} result for all tame topological Collect--Eckmann multimodal maps of an interval, all tame topological Collect--Eckmann rational functions of the Riemann sphere, and all dynamically semi--regular transcendental meromorphic functions from $\mathbb{C}$ to $\widehat{\mathbb{C}}$.<br />Comment: arXiv admin note: text overlap with arXiv:1609.03610
- Subjects :
- Applied Mathematics
010102 general mathematics
Hitting time
General Physics and Astronomy
Hölder condition
Riemann sphere
Statistical and Nonlinear Physics
Interval (mathematics)
Subshift of finite type
01 natural sciences
Measure (mathematics)
010101 applied mathematics
Combinatorics
symbols.namesake
symbols
Ergodic theory
Mathematics - Dynamical Systems
0101 mathematics
Primary 37B20, Secondary 37A25
Mathematical Physics
Meromorphic function
Mathematics
Subjects
Details
- ISSN :
- 13616544 and 09517715
- Volume :
- 33
- Database :
- OpenAIRE
- Journal :
- Nonlinearity
- Accession number :
- edsair.doi.dedup.....d6e663e947996d8e1306fd2b9c89b455