Measure-theoretic degrees and topological pressure for non-expanding transformations
We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equi...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Mihailescu, Eugen [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2014transfer abstract |
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| Schlagwörter: |
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| Umfang: |
23 |
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| Übergeordnetes Werk: |
Enthalten in: Corrigendum to “Rifampicin resistance mutations in the rpoB gene of - Urusova, Darya V. ELSEVIER, 2022, Amsterdam [u.a.] |
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| Übergeordnetes Werk: |
volume:267 ; year:2014 ; number:8 ; day:15 ; month:10 ; pages:2823-2845 ; extent:23 |
| Links: |
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| DOI / URN: |
10.1016/j.jfa.2014.07.026 |
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ELV018053432 |
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| 520 | |a We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. | ||
| 520 | |a We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. | ||
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| 700 | 1 | |a Urbański, Mariusz |4 oth | |
| 773 | 0 | 8 | |i Enthalten in |n Elsevier |a Urusova, Darya V. ELSEVIER |t Corrigendum to “Rifampicin resistance mutations in the rpoB gene of |d 2022 |g Amsterdam [u.a.] |w (DE-627)ELV007566018 |
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10.1016/j.jfa.2014.07.026 doi GBVA2014022000024.pica (DE-627)ELV018053432 (ELSEVIER)S0022-1236(14)00311-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Mihailescu, Eugen verfasserin aut Measure-theoretic degrees and topological pressure for non-expanding transformations 2014transfer abstract 23 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. 37Lxx Elsevier 46G10 Elsevier 46B22 Elsevier 37A35 Elsevier 37D35 Elsevier Urbański, Mariusz oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:267 year:2014 number:8 day:15 month:10 pages:2823-2845 extent:23 https://doi.org/10.1016/j.jfa.2014.07.026 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 267 2014 8 15 1015 2823-2845 23 045F 510 |
| spelling |
10.1016/j.jfa.2014.07.026 doi GBVA2014022000024.pica (DE-627)ELV018053432 (ELSEVIER)S0022-1236(14)00311-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Mihailescu, Eugen verfasserin aut Measure-theoretic degrees and topological pressure for non-expanding transformations 2014transfer abstract 23 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. 37Lxx Elsevier 46G10 Elsevier 46B22 Elsevier 37A35 Elsevier 37D35 Elsevier Urbański, Mariusz oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:267 year:2014 number:8 day:15 month:10 pages:2823-2845 extent:23 https://doi.org/10.1016/j.jfa.2014.07.026 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 267 2014 8 15 1015 2823-2845 23 045F 510 |
| allfields_unstemmed |
10.1016/j.jfa.2014.07.026 doi GBVA2014022000024.pica (DE-627)ELV018053432 (ELSEVIER)S0022-1236(14)00311-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Mihailescu, Eugen verfasserin aut Measure-theoretic degrees and topological pressure for non-expanding transformations 2014transfer abstract 23 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. 37Lxx Elsevier 46G10 Elsevier 46B22 Elsevier 37A35 Elsevier 37D35 Elsevier Urbański, Mariusz oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:267 year:2014 number:8 day:15 month:10 pages:2823-2845 extent:23 https://doi.org/10.1016/j.jfa.2014.07.026 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 267 2014 8 15 1015 2823-2845 23 045F 510 |
| allfieldsGer |
10.1016/j.jfa.2014.07.026 doi GBVA2014022000024.pica (DE-627)ELV018053432 (ELSEVIER)S0022-1236(14)00311-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Mihailescu, Eugen verfasserin aut Measure-theoretic degrees and topological pressure for non-expanding transformations 2014transfer abstract 23 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. 37Lxx Elsevier 46G10 Elsevier 46B22 Elsevier 37A35 Elsevier 37D35 Elsevier Urbański, Mariusz oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:267 year:2014 number:8 day:15 month:10 pages:2823-2845 extent:23 https://doi.org/10.1016/j.jfa.2014.07.026 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 267 2014 8 15 1015 2823-2845 23 045F 510 |
| allfieldsSound |
10.1016/j.jfa.2014.07.026 doi GBVA2014022000024.pica (DE-627)ELV018053432 (ELSEVIER)S0022-1236(14)00311-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Mihailescu, Eugen verfasserin aut Measure-theoretic degrees and topological pressure for non-expanding transformations 2014transfer abstract 23 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. 37Lxx Elsevier 46G10 Elsevier 46B22 Elsevier 37A35 Elsevier 37D35 Elsevier Urbański, Mariusz oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:267 year:2014 number:8 day:15 month:10 pages:2823-2845 extent:23 https://doi.org/10.1016/j.jfa.2014.07.026 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 267 2014 8 15 1015 2823-2845 23 045F 510 |
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Measure-theoretic degrees and topological pressure for non-expanding transformations |
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We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. |
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We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. |
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We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μ ϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree of f | Λ : Λ → Λ , with respect to the measure μ ϕ on the fractal set Λ. In our case, the equilibrium measure μ ϕ is the unique linear functional in C ( Λ ) ⁎ tangent to the pressure function P ( ⋅ ) : C ( Λ ) → R at ϕ. In particular, for the measure of maximal entropy μ 0 of f | Λ , we obtain the asymptotic degree of f | Λ , which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n → ∞ ; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μ ϕ , with respect to arbitrary iterates f m , m ≥ 2 . We then show that a formula for the topological pressure P ( ϕ ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μ ϕ -measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets. |
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